Geometric approach to evolution problems in metric spaces · komen uit de theorie van metrische...

Geometric approach to evolution

problems in metric spaces

Proefschrift

ter verkrijging vande graad van Doctor aan de Universiteit Leiden,

op gezag van de Rector Magnificusprof. mr. P.F. van der Heijden,

volgens besluit van het College voor Promotieste verdedigen op dinsdag 19 april 2011

klokke 15:00 uur

door

Igor Stojkovic

geboren te Belgradoin 1972

Samenstelling van de promotiecommisie

promotor: prof.dr. S.M. Verduyn Lunel

copromotor: dr.ir. O.W. van Gaans

overige leden: prof.dr. L. Ambrosio (Scuola Normale Superiore, Pisa, Italie)prof.dr. J.M.A.M. van Neerven (Technische Universiteit Delft)prof.dr. M.A. Peletier (Technische Universiteit Eindhoven)prof.dr. P. Stevenhagen

Geometric approach to evolution

problems in metric spaces

Research leading to this PhD thesis supported by VIDI grant 639.032.510 of theNetherlands Organisation for Scientific Research (NWO)

Artwork on the cover designed by Brett Daniel, www.BrettDaniel.com

c© Igor Stojkovic, Leiden, 2011

MAJCI, OCU, I SESTRI

Contents

Samenvatting xi

1 Introduction 11.1 Gradient flows, maximal monotone operators and product formulas

in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Gradient flows . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Maximal monotone operators . . . . . . . . . . . . . . . . . 61.1.3 Trotter-Kato product formulas . . . . . . . . . . . . . . . . 8

1.2 The Monge-Kantorovich problem and theWasserstein distances on spaces of probability measures . . . . . . 101.2.1 The Monge problem . . . . . . . . . . . . . . . . . . . . . . 101.2.2 The Kantorovich problem . . . . . . . . . . . . . . . . . . . 121.2.3 Wasserstein distances . . . . . . . . . . . . . . . . . . . . . 13

1.3 Gradient flows in metric spaces . . . . . . . . . . . . . . . . . . . . 141.4 Research topics studied in this thesis . . . . . . . . . . . . . . . . 20

2 Approximation for convex functionals on non-positively curvedspaces and the Trotter-Kato product formula 292.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Construction of approximation semigroups

and some convergence theorems . . . . . . . . . . . . . . . . . . . . 462.4 The Trotter product formula . . . . . . . . . . . . . . . . . . . . . 612.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Wasserstein-2 analysis of the non-symmetric Fokker-Planck equa-tion and the Trotter-Kato Product Formula 733.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3 Construction of the semigroup on (P2,W2) – The Trotter-Kato

product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.4 Absolute continuity of paths and the regularising effect . . . . . . . 93

vii

Contents

3.5 Some remarks about the invariant measure—the symmetric versusthe non-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . 103

4 Maximal Monotone operators in generalised sense on the Wasser-stein space P2(Rd) 1094.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.2 Maximal monotone operators on Hilbert

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3 Ambrosio-Gigli-Savare Frechet subdifferential of geodesically con-

vex functionals on P2(Rd) . . . . . . . . . . . . . . . . . . . . . . . 1164.4 Maximal λ-Monotone Operators in generalized sense, associated

Cauchy problems and the Resolvents on (P2(R2),W2) . . . . . . . 1234.5 AGS Subdifferentials of regular λ-convex functionals in generalized

sense as λ-MMGR operators . . . . . . . . . . . . . . . . . . . . . . 1324.6 Convex subsets of (P2(Rd),W2) in geodesic and in generalized sense 1404.7 Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.8 The abstract Cauchy problem and the construction of the semigroup 155

4.8.1 Uniqueness and the semi-contraction property of the solutions1564.8.2 The exponential formula - Part 1 . . . . . . . . . . . . . . . 1584.8.3 The exponential formula - Part 2 . . . . . . . . . . . . . . . 162

4.9 Towards an application . . . . . . . . . . . . . . . . . . . . . . . . 172

5 Invariant measures for locally Lipschitz stochastic delay equa-tions 1755.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.3 Variation-of-constants formula . . . . . . . . . . . . . . . . . . . . . 1795.4 The equation and the segment process . . . . . . . . . . . . . . . . 1865.5 Tightness of segments . . . . . . . . . . . . . . . . . . . . . . . . . 1915.6 A stability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.7 Markov and eventual Feller property and existence of invariant mea-

sure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Bibliography 207

Curriculum vitae 215

viii

Acknowledgment

First of all, I acknowledge the financial support of the Netherlands Organizationfor Scientific Research (NWO).

I thank my advisors Sjoerd Verduyn Lunel and Onno van Gaans for grantingme freedom in choosing my own research topics. I also thank them for havingconfidence in me and for supporting me morally regarding my decision to stepinto the new field of gradient flows in metric spaces. I specially thank Onno vanGaans for his careful checking of my manuscript and many suggestions regardingimprovement of many of my introductory texts. Onno hes helped me a lot to adda nice touch to my manuscript, which has really made it a better thesis. Contraryto his conviction that he only did job in this regard, I think that some of his inputhas been done with all his hart—which I believe surpasses his job requirements.And I am greatfull for it.

During the last period of preparing my 230 pages thesis a lot of polishing workneeded to be done. At times we worked together on polishing it far beyond theoffice hours. I find it extraordinary kind of him to help me in this way, and I cannot thank him enough for it.

I thank Professer Philippe Clement for his seminars on gradient flows in metricspaces, which have introduced me into this beautiful field. I appreciate that heproposed me a problem which invoked my engagement in this field.

I thank Professor Giuseppe Savare and Professor Luigi Ambrosio for invitingme to their institutes and for interesting discussions from which I learned newthings. I moreover thank Professor Ambrosio for his kind accepting of the taskof being a member of the commission which has approved this thesis, and for hisadvice regarding my publications.

I would like to express my gratitude to Anton Petrunin for his hospitality duringmy visit at University of Munster, and for his generous sharing of his mathematicalinsights which helped me to improve my results of Chapter 2.

I thank Professor Genaro Lopez for a productive visit to his university inSevilla, where I had a very pleasant time.

My mathematics will be appreciated only by the fellow scientists, but I hopethat the artwork on the cover might be enjoyed by many. I thank the author BrattDaniel for making this awesome drawings, and I hope that he will know that I amvery happy with it.

I thank my mother for all her care, understanding and support. Mum, thank

ix

you! Even though I know you never like hearing ’thank you’ from me.Finally, I thank all of my friends who gave me a warm shelter in their harts in

difficult times.

x

Samenvatting

In dit proefschrift presenteert de auteur de resultaten van het onderzoek uitgevoerdtijdens zijn aanstelling als promovendus aan de Universiteit Leiden. Vier essentieelverschillende onderwerpen zijn bestudeerd en de verkregen resultaten kunnen ge-kenmerkt worden als uitbreiding van de bestaande theorie voor gradientstromingenop metrische ruimten. Een belangrijk uitgangspunt zijn gradientstromingen opWasserstein-2 ruimten van kansmaten op Euclidische ruimte Rd.

De theorie van gradientstromingen kan gezien worden als een onderdeel van detheorie van optimaal transport. Deze theorie staat zeer in de belangstelling en inhet afgelopen decennium zijn er een aantal belangrijke doorbraken gerealiseerd.Een belangrijk streven in de studie van de gradientstromingen op verschillendemetrische ruimten, is het vinden van tegenhangers van de elegante theorie van devariationele analyse op lineare ruimten. Een wezenlijk deel van deze tegenhangersis geconstrueerd door Ambrosio-Gigli-Savare in de monografie [5]. In dit werkmaken de auteurs op een essentiele manier gebruik van specifieke meetkundigeeigenschappen van de Wasserstein-2 ruimten over Hilbertruimten, meer specifiekvan de gegeneraliseerde convexiteit van de functie W 2

2 . Een ander belangrijk stre-ven in de theorie van gradientstromingen op metrische ruimten is het vinden vannieuwe toepassingen van de abstracte theorie en, in het bijzonder, het vinden vanpartiele differentiaalvergelijkingen die in zekere zin als dergelijke gradientstrominggeınterpreteerd kunnen worden.

In zijn onderzoek heeft de auteur zijn aandacht gericht op zowel de uitbreidingvan de bestaande abstracte theorie van de gradientstromingen op metrische ruim-ten (in twee verschillende richtingen), alsmede op een behandeling van een klassevan partiele differentiaalvergelijkingen die niet kunnen worden geınterpreteerd alseen gradientstroming op een Wasserstein-2 ruimte, maar wel in een verwante con-text met de Wasserstein-2 ruimte als toestandsruimte.

Een natuurlijk en belangrijk resultaat in de theorie op Hilbertruimten, vanuithet oogpunt van theorie zowel als toepassing, zijn de zogenaamde Trotter-Katoproductformules, die ook wel bekend staan als de splitsingsmethode. De doorTrotter oorspronkelijk bewezen formule luidt:

limn→∞

(e−tnAe−

tnB)n = e−(A+B), ∀t ∈ R, (0.0.1)

voor alle matrices A,B ∈ Rd×d. Het verband tussen formule (0.0.1) en de pro-ductformule op Hilbertruimten, en algemener op de CAT(0)-ruimten, is dat voor

xi

Samenvatting

elke matrix A ∈ Rd×d en voor elke x ∈ Rd de kromme 0 6 t 7→ e−tAx de uniekeoplossing is van de differentiaalvergelijking

ddtx(t) = −Ax(t), t > 0, x(0) := x, (0.0.2)

zodat formule (0.0.1) uigedrukt kan worden in termen van de halfgroepen van deoplossingen van differentiaalvergelijkingen met respectievelijk de matrices A enB als generatoren. Zo’n uitspraak is, zodra twee gradientstromingshalfgroepengegeven zijn, in een algemenere puur metrische context gemakkelijk te formuleren.Het is echter geen eenvoudige zaak of de gegeneraliseerde Trotter-Kato formule inzo’n algemene context nog geldt.

De primaire doelstelling van het onderzoek van de auteur gepresenteerd inHoofdstuk II is om de natuurlijke analogie van de productformules in het kadervan gradientstromen op CAT (0)-ruimten te bewijzen. CAT(0)-ruimten zijn me-trische ruimten met niet-positieve kromming in de zin van Alexandrov. Vanuit nu-merieke approximatie gezien zijn de versies van productformules waar resolventenworden gebruik meer interessant dan die geformuleerd in termen van halfgroepen.Het bewijs van de productformules in Hilbertruimten vereist een aantal andereapproximatiestellingen en specifiek wiskundig gereedschap met betrekking tot in-tegratie van krommen en een ongelijkheid zoals die van Gronwall. Tegenhangersvan deze resultaten zijn in de context van CAT(0)-ruimten ook nodig gebleken—uitgaan van een onderliggende ruimte die niet-lineair is, zorgt er niet voor dathet bewijs meer direct wordt. Technieken die in Hoofdstuk II worden gebruiktkomen uit de theorie van metrische meetkunde, gradientstromingen op metrischeruimten, en evolutie vergelijkingen op Banach ruimten. Gebrek an een bevre-digende analogie van het concept van de zwake convergentie op CAT(0 ruimtenheeft een forse belemering gegeven. Waar in de bewijzen van de productformulesop Hilbertruimten het concept van zwakke convergentie gebruikt wordt, wordenin het algemenere CAT(0)-geval ultra-limieten, ultra-producten en ultra-extensietechnieken successvol gebruikt.

Het centrale probleem dat in Hoofdstuk III wordt beschouwd is de vraag of destroming van de oplossingen van de niet-symmetrische Fokker-Planck vergelijkin-gen

∂tρt = ∆xρt +∇ · (bρt), in D′((0,+∞)× Rd) (0.0.3)

(waar b : Rd → Rd een monotone afbeelding is, maar niet noodzakelijk eengradient van een convexe functie) een contractieve halfgroep op de Wasserstein-2ruimte (P2(Rd),W2) induceert, die bovendien dezelfde padregulariteitseigenschap-pen heeft als de halfgroepen van gradientstromingen (bijvoorbeeld lokaal Lip-schitz in de tijdvariabele). Terwijl de halfgroepen geınduceerd door oplossingenvan de symmetrische versies van de Fokker-Planck vergelijkingen wel gradient-stromingshalfgroepen zijn, kunnen de oplossingen van de niet-symmetrische Fok-ker-Planck vergelijkingen niet geınduceerd worden door een convexe functionaalop (P2(Rd),W2). Deze constatering impliceert dat de bestaande theorie niet al-gemeen genoeg is om de vergelijkingen (0.0.3) te analyseren, althans niet door

xii

middel van rechtstreekse toepassing van de reeds bewezen stellingen. Het onder-zoek naar de niet-symmetrische Fokker-Planck vergelijkingen kwam op gang nadatde auteur had bewezen dat de Trotter-Kato productformule voor de symmetrischeversies van de Fokker-Planck vergelijkingen geldt (dat wil zeggen wanneer b weleen gradient is), ten opzichte van de Wasserstein-2 metriek. Dezelfde methodekan gebruikt worden om de stroming van de oplossingen van de niet-symmetrischeFokker-Planck vergelijkingen ook als een halfgroep op (P2(Rd),W2) te beschou-wen en te bewijzen dat de halfgroep in deze zin contractief is. Verder wordt inHoofdstuk III bewezen dat de paden van deze halfgroep lokaal absoluut continuzijn en zelfs dat het zogenaamde regulariserende effect optreedt. Het geheel van deverkregen resultaten bevestigt dat de meetkundige structuur van de Wasserstein-2ruimten inderdaad goed verenigbaar is met de structuur van de niet-symmetrischeFokker-Planck vergelijking. Anders gezegd, de Waserstein-2 ruimte is inderdaadeen natuurlijke omgeving voor de niet-symmetrische Fokker-Planck vergelijking.Bovendien, als men de hoodfstelling van Hoofdstuk III met de hoodfstelling overde gradientstromingen (toegepast op de symmetrische Fokker-Planck vergelijking)vergelijkt, kan men een analogie vaststellen met de verhouding in Hilbertruim-ten van de gradientstromingen tot de stromingen die geınduceerd worden doormaximale monotone operatoren.

Hoofdstuk IV besteedt weer aandacht aan het uitbreiden van de abstracte the-orie (althans in eerste instantie), en het doel van de auteur is om een bevredigendetheorie van maximale monotone operatoren in een passende gegeneraliseerde zinop te bouwen. Een essentieel uitgangspunt is dat de nieuwe theorie een uitbreidingdient te zijn van de theorie van de gradientstromingen op Wasserstein-2 ruimten.Een stroming wordt een gradientstroming genoemd als deze wordt geınduceerddoor functionalen die onderhalfcontinu zijn en convex langs gegeneraliseerde ge-odeten. Een basisuitgangspunt van dit onderzoek is dat men gebruik kan makenvan de gegeneraliseerde convexiteit van de functie W 2

2 , dezelfde meetkundige ei-genschap van (P2(Rd),W2) die door de auteurs van [5] op essentiele wijze gebruiktis in de opbouw van hun theorie.

Hoofdstuk IV is als volgt opgebouwd. Eerst wordt in gegeneraliseerde zin hetbegrip maximale monotone operatoren op Wasserstein-2 ruimten geıntroduceerd.Vervolgens wordt aangetoond dat de Frechet subdifferentiaal, zoals gedefinieerddoor Ambrosio-Gigli-Savare voor functionalen die convex zijn langs de gegenera-liseerde geodeten, maximale monotone operatoren volgens deze definitie zijn. Hetblijkt nu dat dergelijke operatoren en de bijbehorende resolventen een aantal ei-genschappen hebben die geheel analoog zijn aan het Hilbertruimte geval. Dezeresultaten gelden ook voor gradientstromingen, maar waren daarvoor niet eer-der in de literatuur bewezen noch geformuleerd. Het onderzoek gepresenteerd inHoofdstuk IV wordt afgerond met het geven van een bewijs van de hoofdstel-ling over de existentie van oplossingen van het bijbehorende Cauchy-probleem ende fundamentele eigenschappen van de geınduceerde halfgroep. Het kan wordenopgemerkt dat er een conceptueel verschil is tussen ons bewijs en de reeds gepu-bliceerde bewijzen van de stellingen over existentie van oplossingen in het kadervan gradientstromingen. Het hoofdstuk wordt afgesloten met een discussie over

xiii

Samenvatting

een mogelijke uitbreiding van onze theorie in het kader van de Wasserstein-2 ruim-ten over oneindig-dimensionale ruimten (bijvoorbeeld de Cameron-Martin ruimtenvan Gaussmaten op Banachruimten) en het behandelen van oneindig-dimensionalewarmtevergelijkingen als toepassing van deze nieuwe theorie. Verder merkt de au-teur op dat de resultaten van de Hoofdstukken III en IV, in aanvulling op deeerder ontwikkelde theorie van Waserstein-2-ruimten, een grondig onderzoek naarde volgende vraag motiveren: Is er een passende Hille-Yosida stelling te bewijzenin het kader van contractiehalfgroepen op de Wasserstein-2 ruimten?

In het laatste Hoofdstuk V wordt onderzoek gepresenteerd waarmee existentievan een invariante maat voor een stochastische differentiaalvergelijking met eentijdvertraging in zowel de drift als in de diffusieterm kan worden bewezen. Dedrift van de stochastische vergelijking is verondersteld exponentieel stabiel te zijn,en het diffusieproces is een Levyproces waar grote sprongen niet al te vaak optre-den. De belangrijkste bijdrage van dit onderzoek vergeleken met de resultaten in[92], is dat de globale Lipschitzvoorwaarde ten aanzien van de diffusiecoefficientversoepeld is tot een lokale Lipschitzvoorwaarde. De zogenaamde variatie-van-constanten formule is een belangrijk en onmisbaar gereedschap in dit onderzoeken deze formule wordt dan ook bewezen in Hoofdstuk V. Verder wordt een stel-ling over stabiliteit van de oplossing ten opzichte van de beginvoorwaarde bewezendie ook onafhankelijk van de andere resultaten van belang is. Dit onderzoek isuitgevoerd tijdens de eerste periode van het promotieonderzoek van de auteur,en daarna gebruikt ter voorbereiding van een publicatie in samenwerking met debegeleider Dr. Onno van Gaans. De auteur heeft een bewuste keuze gemaakt omdeze resultaaten in het laaste hoofdstuk te presenteren, aangezien de behandeldeonderwerp tot de theorie van de stochastische analyse behoort, en niet tot de the-orie van optimale transport en gradientstromen, het centrale onderwerp van ditproefschrift.

xiv

Chapter 1

Introduction

1.1 Gradient flows, maximal monotone operatorsand product formulas in Hilbert spaces

1.1.1 Gradient flows

Partial differential equations (PDE’s) are undoubtedly the most commonly usedmathematical model for various physical phenomena. A gradient flow system isa particular type of a differential equation. Its study has a long history andsuch systems are now well understood. More recently it has been discovered thatmany other PDE’s can be viewed as gradient flows if one generalizes this conceptto a suitable more abstract setting. In this section we give a brief overview ofthe classical theory of gradient flows on Hilbert spaces, and also of the theory ofmaximal monotone operators on Hilbert spaces, which is a natural extension ofthe theory of gradient flows. To fix the ideas, let us first define gradient flowson Rd. Let ϕ : Rd → R be a continuously differentiable function. Consider thefollowing problem:

ddtx(t) = −∇ϕ(x(t)), t > 0,

x(0) = x0 ∈ Rd,(1.1.1)

where x : [0,+∞)→ Rd is the unknown function to be solved from the equation.Its initial value at time t = 0 is an arbitrary but fixed point x0 ∈ Rd. Due to ourassumption that ϕ be continuously differentiable, one easily argues by means ofPicard iteration that (1.1.1) has a unique solution for each x0 ∈ Rd. The functionϕ is often called the potential of the gradient flow equation (1.1.1), solutions areusually called gradient flow curves, and the set of all solutions defines the gradientflow associated to the potential ϕ.

1

CHAPTER 1: Introduction

Approximation scheme

The most suitable method for computing the approximate values of the solutionsturns out to be the Euler method, i.e. given the initial value x(0) = x0 ∈ Rd, anda time step size τ > 0, one makes the time scale discrete by restricting the set ofobservations to times 0, τ, 2τ, 3τ, ... and ‘computes’ the approximate value x(τ) by

x(τ)− x(0)τ

≈ −∇ϕ(x(τ)) (1.1.2)

(which replaces the differential equation (1.1.1)). In (1.1.2) x(τ) is the unknownvalue implicitly determined by this equation, and rewriting gives

x(τ) = (I + τ∇ϕ)−1x(0), (1.1.3)

where I : Rd → Rd is the identity map. Naturally, the operator (I + τ∇ϕ)−1 :Rd → Rd in (1.1.3) must be well defined for each τ > 0, and it turns out that thenecessary and sufficient condition for this to hold is that ϕ : Rd → Rd is a convexfunction, i.e.

ϕ((1− t)x+ ty) 6 (1− t)ϕ(x) + tϕ(y), ∀x, y ∈ Rd,∀ t ∈ [0, 1]. (1.1.4)

Since convexity of ϕ implies monotonicity of its gradient ∇ϕ, that is

〈∇ϕ(x)−∇ϕ(y), x− y〉 > 0, ∀x, y ∈ Rd, (1.1.5)

the so called resolvents associated to ∇ϕ, i.e.

Jτ := (I + τ∇ϕ)−1 : Rd → Rd, ∀τ > 0, (1.1.6)

are well defined. This claim can be seen to hold as follows. A differentiable functionassumes a local extremal value at a point x ∈ Rd, if and only if its gradient is 0at x. Therefore, for each x ∈ Rd and τ > 0, Jτx must be the unique minimizer ofthe function

Rd 3 y 7→ Φτ (x, y) :=12τ|x− y|2 + ϕ(y), (1.1.7)

since∇yΦτ (x, y) = 1τ (y−x)+∇ϕ(y). Conversely, any minimizer xτ of the function

Φτ (x, y) (over y ∈ Rd) solves the implicit equation posed in (1.1.2). With the aidof the convexity assumption on ϕ, it can be easily proven that this minimizationproblem has a unique solution, hence the operators Jτ in (1.1.6) are well defined foreach τ > 0. The approximate value x(2τ) of the solutions of (1.1.1) is defined byx(2τ) := (I + τ∇ϕ)−1x(τ) = Jτ x(τ) = J2

τ x(0), and the values of the approximatesolution at the subsequent times 3τ, 4τ, ... are given by

x(kτ) := (I + τ∇ϕ)−1x((k − 1)τ) = Jτ x((k − 1)τ) = Jkτ x(0), k ∈ N. (1.1.8)

In order to stress the dependence of the approximate solution on the time stepsize τ in the approximation, we write xτ (kτ), k ∈ N.

2

Section 1.1

At this point in our analysis several considerations should be given.The first consideration regards the accuracy of the above discussed approxi-

mation procedure. More precisely, if we let τ converge to 0, do our approximatediscrete time solutions converge in some way to the unique solution of the abstractCauchy problem (1.1.1), and if they do what is the order of this convergence? Theanswer to this question is indeed affirmative, i.e. for each t > 0 and for eachx0 ∈ Rd, we have that (

J tn

)nx0 −→ x(t), as n→ +∞, (1.1.9)

where 0 6 t 7→ x(t) is the unique solution of (1.1.1), and the order of this conver-gence is 1

n .The second consideration regards relaxing our assumption that ϕ : Rd → R

is continuously differentiable, to a weaker asumption. In this case we also needto reformulate the Cauchy problem (1.1.1) since the gradient of ϕ appears in itexplicitly.

Subdifferential of a convex function

It turns out that a sufficient pair of conditions for the above described procedureto be implemented is that ϕ be convex (see (1.1.4)) and lower semi-continuous,i.e.

ϕ(x) 6 lim infn→∞

ϕ(xn) whenever xn → x, (1.1.10)

holds. The gradient ∇ϕ of a C1 function ϕ then needs to be replaced by the socalled subdifferential ∂ϕ of ϕ, which due to the assumed convexity can be definedby

Rd × Rd ⊃ ∂ϕ := (x, ξ)|〈ξ, y − x〉+ ϕ(x) 6 ϕ(y) ∀y ∈ Rd. (1.1.11)

It should be noticed that ∂ϕ is in general a multi-valued operator, i.e. there maybe points x such that ∂ϕ(x) contains more than one point—in fact we have defined∂ϕ to be a relation rather than a function. An easy example when this occurs isobtained by taking d := 1, ϕ := (R 3 x 7→ |x|), and observing that ∂ϕ(0) = [0, 1],an uncountable set. In terms of high school calculus, elements of ∂ϕ(x) are inone-to-one correspondence with all hyperplanes in Rd=1 that touch the graph ofϕ from ‘below’ (infinitesimally near x even when ϕ is not convex). In this moregeneral context, the equation (1.1.1) is replaced by the following abstract Cauchyproblem:

ddtx(t) ∈ −∂ϕ(x(t)), for L1-a.e. t > 0,

x(0) := x0 ∈ D(∂ϕ),(1.1.12)

where D(∂ϕ) denotes the closure of the domain of ∂ϕ. Moreover, one may considerfunctions ϕ : Rd → (−∞,+∞] (i.e. ϕ may assume values +∞). In such case thegradient flow of the solutions to (1.1.1) is defined on the closure of the properdomain D(ϕ) := x ∈ Rd|ϕ(x) <∞ of ϕ.

3


Gradient flows in Hilbert spaces

Our next consideration is whether we really need to work in Rd, or can we extendour results to infinite dimensional Hilbert or perhaps Banach spaces. It turns outthat the theory of gradient flows on Rd is easily extended to arbitrary Hilbertspaces, while any other Banach space setting does not work in general, and inorder to insure existence of the discrete approximate solutions, as well as theproper solutions in continuous time one typically assumes the lower level sets of ϕto be relatively compact. The reason that there is no general theory of gradientflows in Banach spaces is simply because the geometry of these spaces does notsupport such constructions, i.e. suitable variational estimates do not hold in suchspaces.

α-convex functionals

Furthermore, the convexity assumption of ϕ can be relaxed to the so called semi-convexity assumption, i.e. the assumption that for some α ∈ R the function

ϕ(α)(x) := ϕ(x)− α

2|x|2 (1.1.13)

is convex suffices, in which case one says that ϕ is α-convex. Since x 7→ |x|22

is convex, while sums of convex functions are also convex, it clearly holds thatfor each α ∈ R any α-convex function is β-convex for any β < α. So α-convexfunctions with α > 0 stay convex even if we subtract a positive multiple (6 α) ofx 7→ |x|2

2 , while α-convex functions with α < 0 actually need to be added a positive

multiple (> α) of x 7→ |x|22 in order to become convex. This technical relaxation

of the assumptions on the potential ϕ turned out very useful for treating variouspartial differential equations.

The main flow generation and properties theorem reads:

Theorem 1.1.1. Let H be a Hilbert space and let ϕ : H → (−∞,+∞] be a lowersemi–continuous function, which is moreover α–convex for some α ∈ R. Then

1. For each x ∈ H and h > 0 such that 1 + αh > 0 the resolvent operator

Jh := (I + h∂ϕ)−1 (1.1.14)

is well defined on H as the unique minimizer of the function

Rd 3 y 7→ Φh(x, y) :=1

2h|x− y|2 + ϕ(y). (1.1.15)

2. For each t > 0 and for each x0 ∈ D(ϕ) the limit

Stx0 := limn→+∞

(J tn

)nx0 (1.1.16)

exists.

4

Section 1.1

3. For each x0 ∈ D(ϕ) the curve t 7→ Stx0 is the unique solution of the abstractCauchy problem

ddtStx0 ∈ −∂ϕ(Stx0), for L1-a.e. t > 0, (1.1.17)

with initial value x0.

4. The mapping [0,+∞)× x ∈ H|f(x) < +∞ 3 (t, x) 7→ Stx is a semigroup,i.e.

S0 = I, St Ss = St+s, ∀ t, s > 0, (1.1.18)

and it is moreover α-contracting, i.e.

|Stx− Sty| 6 e−αt|x− y|, ∀x, y ∈ D(ϕ), ∀ t > 0. (1.1.19)

5. For each x0 ∈ D(ϕ) the curve 0 6 t 7→ Stx0 is Lipschitz on each compactsubinterval in (0,+∞). Moreover, if x0 ∈ D(∂ϕ), then the convergence in(1.1.16) is of order 1

n .

In addition, for each x0 ∈ D(ϕ), we have that ϕ(Stx0) < +∞ for eacht > 0, and functions 0 < t 7→ e−αtϕ(Stx0), and 0 < t 7→ e−αt

∣∣ ddtStx0

∣∣ arenon-increasing.

The theory discussed above is usually referred to as the theory of gradientflows on Hilbert spaces, and it is considered to be a part of the larger theory ofvariational analysis, also called the calculus of variations theory.

A fundamental example: the heat equation on Rd

Let us give an important example of a partial differential equation that can beinterpreted as a gradient flow in a suitable Hilbert space. Recall the heat equationon Rd, i.e.

ddtu(t, x) = ∆u(t, x) =

d∑j=1

∂j∂ju(t, x), (1.1.20)

and let H := L2(Rd,dx). Recall moreover the Sobolev space W 1,2(Rd) of squareintegrable functions on Rd, whose first order distributional derivatives are alsosquare integrable functions. Define the following functional on L2(Rd,dx)

ϕ(u) :=

∫Rd |∇u|

2dx if u ∈W 1,2(Rd)+∞ otherwise.

(1.1.21)

It is well known that for each u ∈ L2(Rd,dx), we have that (u,∆u) ∈ −∂ϕ ifu ∈ W 1,2(Rd) and ∆u ∈ L2(Rd,dx), and that the gradient flow associated to ϕ(see Theorem 1.1) gives the L2 solutions of equation (1.1.20). The heat equationhas been studied in many different contexts and the variational method is oneway to look at it. A key observation which has set the foundations of variational

5


analysis is that solutions of certain differential equations are minimizers of anappropriate functional defined on some linear space of functions where one expectsto find solutions—the partial differential equation in question is interpreted as anoperator on this linear space of functions, and if this operator coincides with thegradient (or more generally with the subdifferential) of a functional ϕ, then anylocally minimizing function f of ϕ satisfies ∇ϕ(f) = 0, which thus amounts to fsolving the partial differential equation under consideration. For example, localminimizers of the functional ϕ defined in (1.1.21) are solutions of the Laplaceequation

∆u =d∑j=1

∂j∂ju = 0. (1.1.22)

The techniques of the calculus of variations have turned out to be very powerfulin theoretical and applied mathematics and physics, as well as in various practicalapplications beyond the scope of these two disciplines.

1.1.2 Maximal monotone operators

There is a natural generalization of the theory of gradient flows on Hilbert spaces.The potential ϕ of the equation (1.1.12) is required only through its subdifferential∂ϕ, and one may be tempted to examine whether more general operators sufficefor proving the existence and uniqueness of solutions of the associated Cauchyproblem. Thus the question reads: are there reasonable operators A ⊂ H × Hsuch that the Cauchy problem

ddtx(t) ∈ −Ax(t), for L1-a.e. t > 0,

x(0) = x0 ∈ D(A),(1.1.23)

has a unique solution for each x0 ∈ D(A). In this, the domain D(A) of A is definedby D(A) := x ∈ H| ∃ξ ∈ H such that [x, ξ] ∈ A. The reader may observe thatthe operator A is in fact defined as a relation contained in H × H, hence maybe multi-valued. In the course of study of various PDE’s, it has been discoveredthat allowing ‘operators’ to be multi-valued has definite advantages, and an im-portant class of examples are the subdifferentials of convex, lower semi-continuousfunctionals defined on Hilbert spaces. Therefore, operators are typically definedas relations contained in H ×H, or as mappings H → 2H . It turned out that thenatural set of conditions that guarantee existence and uniqueness of solutions of(1.1.23) producing a generalization of the theory of gradient flows, are the so calledmaximal monotonicity conditions. Precisely, a subset A ⊂ H × H is a maximalmonotone operator (H is a given Hilbert space), if the following two conditionshold

1. A is a monotone operator, i.e. for each [x1, ξ1], [x2, ξ2] ∈ A we have that

〈x1 − x2, ξ1 − ξ2〉 > 0 (1.1.24)

6

Section 1.1

2. A is a maximal element in the class of monotone subsets of H × H withrespect to the set inclusion, i.e. for any monotone subset B ⊂ H × H,A ⊂ B implies A = B.

It can be shown that a monotone subset A ⊂ H ×H is maximal if and only if foreach h > 0 the associated resolvent operator

Jh := (I + hA)−1 ⊂ H ×H (1.1.25)

is single valued and defined on H. As we already explained, subdifferentials ofconvex functionals are maximal monotone operators. More generally, one canconsider maximal α-monotone operators for α ∈ R, i.e. operators A ⊂ H × Hfor which the operator A − αI is a maximal monotone operator. Clearly, thesubdifferential of an α-convex functional defined on H is maximal α-monotone.Similar to the case of gradient flows, the resolvents Jh defined in (1.1.25) can beused to construct a ‘numerical approximation’ of the solutions of (1.1.23), the limit

Stx0 =(J tn

)nx0 exists, and the curve 0 6 t 7→ x(t) := Stx0 is the unique solution,

for each initial value x0 ∈ D(A). Furthermore, (St)t>0 is a semigroup with thesame kind of properties as stated in Theorem 1.1.1, except for one structuraldifference. In the gradient flow case, for each point x0 ∈ D(ϕ) = D(∂ϕ) (theequality here is not hard to prove), the curve 0 6 t 7→ Stx0 is Lipschitz on eachcompact subinterval of (0,+∞), it solves the Cauchy problem (1.1.23) on (0,+∞),and for any t > 0 we have that Stx0 ∈ D(∂ϕ). However, in the general case ofmaximal monotone operators, the semigroup of solutions does not possess sucha regularizing effect, and only the paths emanating from points x0 ∈ D(A) aresolutions of (1.1.23) with the local Lipschitz property.

A well known example of an equation that can be interpreted as a Cauchyproblem with a maximal monotone operator, is the wave equation

∂ttu(t, x) = ∆u(t, x),u(0, x) = f(x),

∂tu(0, x) = g(x),(1.1.26)

where f and g are given functions defined on Rd. In order to apply the theoryof maximal monotone operators, one interprets (1.1.26) as the ’two dimensional’system of equations

∂tu = v,

∂tv = ∆u,u(0, ·) = f,

v(0, ·) = g.

(1.1.27)

Now choose H := L2(Rd,dx)×L2(Rd,dx), and define the operator A ⊂ H×H,A(u, v) := −(v,∆u), D(A) := D(∂ϕ)×L2(Rd,dx), where ϕ denotes the functionaldefined in (1.1.21). It can be shown that A is a maximal monotone operator and

7


one easily sees that (1.1.27) amounts to

ddt

(u(t), v(t)) ∈ −A(u(t), v(t)),

(u(0, ·), v(0, ·)) =(f, g).(1.1.28)

1.1.3 Trotter-Kato product formulas

An integral part of the theory of gradient flows and more generally of maximalα-monotone operators on Hilbert spaces, are various approximation theorems, andin particular the Trotter-Kato product formula. In order to keep the expositionsimple, let us consider a Hilbert space H and two convex lower semi–continuousfunctionals

ϕ1, ϕ2 : H → (−∞,+∞] (1.1.29)

such that the sum functional ϕ := ϕ1 + ϕ2 has non-empty domain, i.e. ϕ 6≡ +∞.it is easy to see that ϕ is convex and lower semi-continuous as ϕ1 and ϕ2 aresuch, hence according to Theorem 1.1.1, each of the functionals ϕ1, ϕ2, and ϕ,induce gradient flow semigroups, which we denote by (S1

t )t>0, (S2t )t>0, and (St)t>0,

respectively. Denote moreover J1h, h > 0, and J2

h, h > 0, to be the resolventoperators associated to ϕ1 and ϕ2, respectively. One version of the Trotter-Katoproduct formula in this context reads:(

J2tnJ1tn

)nx −→ Stx, as n→ +∞, t > 0, (1.1.30)

for each x ∈ D(ϕ). Moreover, this convergence is uniform on each compact timeinterval. Furthermore, denoting P 1 and P 2 to be the nearest point projectionsonto the closed convex subset E1 := D(ϕ1) and E2 := D(ϕ2), respectively, wehave the following version of the product formula:((

S2tn P 2

)(S1tn P 1

))nx −→ Stx, as n→ +∞, t > 0, (1.1.31)

for each x ∈ D(ϕ), and this convergence is uniform on compact time intervals.More generally, one can prove the product formulas associated to any finite

number of convex functionals with mixed versions of the product formula wheresome steps in de approximation procedure may be given by the correspondingsemigroup and others by the resolvent. Furthermore, product formulas associatedto sums of pairs (or more generally of finite sequences) of maximal monotoneoperators can be proven, too, provided that the sum operator is also maximalmonotone.

Besides for being a natural and theoretically fundamental result, product for-mulas have proven to be very useful for treating various concrete problems. Forexample, there are differential equations where several functionals (or operators)contribute to the evolution of the system, where each of them separately beingwell handleable, but such that the interaction of the various factors is quite hard,

8

Section 1.1

or even impossible to handle. Let a convex functional ϕ : H → (−∞,+∞] begiven and suppose that we need to restrict the evolution of the system to a convexsubset B ⊂ H. This can be modelled by adding the ‘indicator’ functional

ψB(x) :=

1 if x ∈ B+∞ if x ∈ H \B,

(1.1.32)

to ϕ, i.e. one considers the gradient flow associated to the functional ϕ1 := ϕ +ψB . In general, it is very hard to say anything about the evolution of such asystem directly (the main difficulty is to determine the evolution after the ‘particle’has hit the boundary of B), but with the aid of the product formulas one canactually do a very detailed analysis. Furthermore, product formulas can be usedto actually construct ‘sums’ of semigroups and show that these ‘sums’ possesscertain properties.

Judging by the classical results mentioned above, product formulas present anatural and useful addendum to the basic theory of gradient flows. Therefore,whenever a mathematical theory emerges which resembles the theory of gradientflows on Hilbert spaces to a substantial degree, one should provide an appropriateapproximation theory as well.

The first product formula for matrices was proved in 1959 by Trotter. For anytwo matrices A,B ∈ Rd×d, Trotter showed that(

e1nAe

1nB)n n→∞−→ e(A+B). (1.1.33)

Observe that any matrix C ∈ Rd×d is the infinitesimal generator of the semigroupof solutions of the ODE system

ddtx(t) = Cx(t), (1.1.34)

whose semigroup of solutions is defined by

SCt x := etCx, t > 0, x ∈ Rd. (1.1.35)

This observation naturally leads one to reformulate the Trotter product formula(1.1.33) in terms of semigroups (SAt )t>0 and (SBt )t>0 (defined by (1.1.35) choos-ing C := A and C := B, respectively), which can be formulated in exactly thesame way for linear C0-semigroups on Banach spaces. The product formula forC0-semigroups on Banach spaces was proven in 1960’s. In light of the classicaldevelopment of the theory of non-linear semigroups induced by accreative oper-ators1, it seems reasonable to attempt proving product formulas for such semi-groups. However, the geometry of general Banach spaces is simply too wild forthis claim to be true in general, and the most general results within the scope of

1We will not give the definition of accretive operators on Banach spaces in this thesis, but weremark that in any Hilbert space accretive operators coincide with the monotone operators

9


linear spaces are product formulas for flows induced by maximal monotone oper-ators on Hilbert spaces. Proofs of product formulas for flows on Hilbert spacesinduced by maximal monotone operators can be found in [17] (see Proposition4.3 and Proposition 4.4 there). Furthermore, in 1978 Kato and Masuda havepublished a proof of the product formula for arbitrary finite sequences of convexlower semi-continuous functionals on Hilbert spaces, where the authors apply avariational calculus method. An extension of the general theory of product for-mulas to semigroups on non-linear spaces is given in Chapter II of this thesis,and to the author’s knowledge it is the first extension at a considerable level ofgenerality beyond the seting of linear spaces. It will be proved that the productformulas associated to finite sequences of geodesically convex functionals definedon complete CAT(0) spaces hold. Since closed convex subsets of Hilbert spacesare complete CAT(0) spaces, our result is en extension of the classical result in[58] by Kato-Masuda.

1.2 The Monge-Kantorovich problem and theWasserstein distances on spaces of probabilitymeasures

During the past decade, the mathematicians community has witnessed a vastdevelopment of a new branch of mathematics called the optimal transportationtheory. In this new multidisciplinary field various techniques which descend fromclassical mathematical disciplines such as calculus of variations, metric geometryand probability theory are jointly applied in order to generate new insights.

This interplay of different disciplines has realized a wide range of applicationswithin mathematics (such as the theory of partial differential equations and thetheory of gradient flows on metric spaces) as well as beyond mathematics (suchas in image processing, urban planning and even in medical science). It would behard to give an accurate list of all the monographs that have been published on thistopic, for many fellow mathematicians have contributed—optimal transportationtheory is certainly one of the major developments in the field of mathematics ofthe first decade of the 21st century. The most thorough exposition of the optimaltransportation theory available at the preset time, is the book [106] written by therecent Fields medal winner Cedric Villani.

1.2.1 The Monge problem

The origin of the optimal transportation theory are the so called Monge problem,and its relaxed version the Kantorovich problem. Even though the Monge problemhas already been posed in 1781 (see [76]), a satisfactory solution has been givenonly in recent years. Let us now pose the Monge problem. Suppose that we havea pile of sand at some location and at another location a hole, such that the pileof sand precisely fits in the hole. Assume moreover that we need to fill the hole

10

Section 1.2

with the sand and that we have to make a certain effort in order to move thesand, say from each point in the pile to each point in the hole, per each unit ofmass. The question is: what is the minimal effort that we have to make in order tomove the sand? A related problem is the following. A producer of certain goodshas a number of factories where the goods are produced (say k), and a numberof locations (stores) where these goods are sold. The producer needs to transportthe goods from his factories to his stores and he must pay a certain price per unitof goods for the transport from each factory to each store. The question reads:what is the minimal cost to transport the goods?

In order to start an analysis of the above described problems, a rigorous math-ematical formulation is needed. Let metric spaces X and Y be given and let µand ν be probability measures defined on the Borel σ-algebras B(X) and B(Y )of X and Y , respectively. Suppose moreover that c : X × Y → [0,+∞] is aB(X)×B(Y ) \B([0,+∞]) measurable (cost) function. The problem reads: mini-mize the expression

I(r) :=∫X

c(x, r(x))dµ(x). (1.2.1)

The choice of the function r : X → Y in (1.2.1) ranges among all admissiblefunctions, i.e. measurable functions which map X into Y and push the probabilitymeasure µ to the probability measure ν, which is denoted ν = r#µ. By definition,ν = r#µ if and only if

ν(B) = µ(r−1(B)), ∀B ∈ B(Y ). (1.2.2)

Notice that if Y = Rd (for some d ∈ N), then ν = r#µ means that the law of runder µ is ν. Obviously, for each admissible function r : X → Y , the expressionI(r) in (1.2.1) is non-negative, hence the infimum over all such expressions is non-negative (if there are no admissible functions r : X → Y , then this infimum equals+∞). Solving this problem is by definition finding a minimizer, i.e. an admissiblefunction which realizes the infimum in (1.2.1), and such minimizer functions aresaid to be solutions of the Monge problem associated to X,µ, Y, ν and the costfunction c. Some basic examples of a Monge problem are obtained by takingX = Y = Rd, c(x, y) := |x − y|p, for all x, y ∈ Rd, with p ∈ [1,+∞). A case ofparticular relevance for applications in PDE’s and gradient flows theory is wherep = 2. One may observe that the examples of the sand pile and the transportationof goods present rather specific situations, from the point of view of the generalformulation. In the sand pile example, both metric spaces X (the location of thesand pile) and Y (the location of the hole) are in fact subsets of the 3-dimensionalEuclidean space, and both probability measures µ and ν are in fact suggested tobe uniform measures on X and Y—sand should have more or less the same densityeverywhere in the pile and hole. In the example of the transport of goods, bothspaces X (the factories) and Y (the stores) are metric spaces consisting of finitelymany points.

The Monge problem may fail to have solutions even in fairly non-pathologicalcases, regarding the data X,Y, µ, ν, c. To illustrate the difficulties which one may

11


encounter, let X = Y = Rd, c(x, y) = |x − y|2, and let µ and ν be probabilitymeasures on Rd with density functions f and g respectively, i.e. dµ = fdx,dν = gdx. Then due to the change of variable formula, any C1 diffeomorphismr : Rd → Rd that satisfies r#µ = ν must also satisfy the following fully non-linearpartial differential equation:

f(x) = g(r(x))|detDr(x)|, (1.2.3)

where Dr denotes the Jacobian of r. Clearly, finding any admissible functions isnot an easy task, even in such a ‘nice’ situation.

1.2.2 The Kantorovich problem

The Monge problem has a relaxed version called the Kantorovich problem. Therelaxed problem is in fact a linear problem on a space of measures and it is mucheasier to solve. Given separable metric spaces X and Y , probability measures µand ν defined on the Borel σ-algebras of X and Y , respectively, and a B(X) ×B(Y ) \ B([0,+∞]) measurable cost function c : X × Y → [0,+∞], minimize thefollowing functional

T (σ) :=∫c(x, y) dσ(x, y). (1.2.4)

Here σ ranges over all admissible probability measures σ ∈ P(X × Y ) (P(X × Y )denotes the set of probability measures on B(X)× B(Y )), i.e.

(πX)#σ = µ, (πY )#σ = ν, (1.2.5)

where the projections πX : X × Y → X and πY : X × Y → Y are defined byπX(x, y) = x, πY (x, y) = y for x ∈ X, y ∈ Y . Thus a minimizer is a probabilitymeasure on B(X)× B(Y ), and any such minimizer is said to be a solution of theKantorovich problem associated to the data X,Y, µ, ν, c.

The Kantorovich problem is a relaxed version of the Monge problem in thefollowing sense. Suppose that r : X → Y is an admissible function for the Mongeproblem. Defining (iX , r) : X → X × Y , (iX , r)(x) := (x, r(x)), one easily seesthat the probability measure σ := (iX , r)#µ is an admissible measure for the Kan-torovich problem (by definition iX(x) = x for each x ∈ X). Informally speakingthe Kantorovich problem is a relaxed version of the Monge problem, since theKantorovich formulation allows splitting of the mass (from the departure spaceX), contrary to the Monge formulation. It is now easily seen that given any dataX,Y, µ, ν, c, the infimum in the Kantorovich problem is less or equal the infimumin the Monge problem—in particular a solution of the Monge problem (with givendata X,Y, µ, ν, c) may not generate a solution of the Kantorovich problem. Thereader may now indeed observe that the Kantorovich problem is a linear problemin the linear space of measures, or rather its convex subset of probability measureson B(X)× B(Y ). Kantorovich won the Nobel price in 1940’s for his related workin the field of economy.

12

Section 1.2

1.2.3 Wasserstein distances

A particularly interesting case occurs when one considers a Polish space (X, d),and the Kantorovich problem with data Y = X, µ ∈ B(X), ν ∈ B(X), andc(x, y) := dp(x, y), x, y ∈ X, with p ∈ [0,+∞). In this case, the Kantorovichproblem always has solutions (by means of weak compactness and convexity argu-ments), and the set of solutions is called the set of optimal admissible transporta-tion plans, denoted Γo(µ, µ) ⊂ P(X ×X) (in this notation the exponent p is notaccounted). Furthermore, the set of all admissible transportation plans is usuallydenoted Γ(µ, ν), and we always have that Γo(µ, ν) ⊂ Γ(µ, ν).

It is not hard to show that the Kantorovich problem (still taking c := d)induces a distance on certain subsets of the set of probability measures P(X), forany separable metric space X. These distances are (to some extent erroneously)called the Wasserstein p distances, denoted Wp, and they are given by choosingX = Y and c(x, y) := dp(x, y) in the Kantorovich problem (d denotes the distancefunction on our separable metric space X, and p ∈ [1,+∞)), and taking the p-throot of the infimum of the functional T over all σ ∈ Γ(µ, ν). This infimum is then aminimum, but the set of minimizers is in general not a one point set. As a mattera fact, one easily sees that the set of minimizers is a convex subset of P(X) (withrespect to the usual linear structure of the set of Borel measures on X), hence oneeither has one minimizer, or infinitely many.

In order to insure that each pair of measures is at finite Wasserstein-p distanceaway from each other, one considers the set Pp(X) of measures µ ∈ P(X) whichhave the property that for some (hence any) point y ∈ X we have that∫

X

dp(x, y) dµ(x) < +∞. (1.2.6)

Alternatively, one may choose to work with a pseudo-distance, i.e. allow that thedistance function assumes the value +∞. Such an approach has turned out usefulif one considers the Wasserstein distances on the set probability measures on theWiener space (see [41], for some fundamental pioneering work on this topic).

Wasserstein spaces and in particular the quadratic case p = 2 have turned outto be very useful in studying the geometry of the space X itself. They have alsofound many applications in the field of PDE’s and gradient flows on metric spaces.An interested reader may consult [15], [42] for some fundamental results on theexistence of solutions of the Monge problem in Euclidean spaces and on manifolds,respectively, while in [41] a fundamental pioneering work on the Monge problem inthe infinite dimensional setting can be found. The list of works where the optimaltransportation theory and the Wasserstein distances are applied to treat PDE’swould probably be too long to fit in this thesis. However, the following works areconsidered to be milestones in this genesis: [16], [54], [85], [22], [5], [21]. Strikingresults in this direction are by Lott-Villani [68], and by Sturm in [102] and [103],where the authors introduce a notion of metric spaces with the Ricci curvaturebounded from below by some K ∈ R. These works are inspired by the fact thaton a Riemannian manifold the lower Ricci curvature is bounded by K if and only

13


if the relative entropy functional with respect to the Riemannian volume measureis K-convex.

1.3 Gradient flows in metric spaces

The classical theory of gradient flows on Hilbert spaces, which is briefly sketchedin Section 1.1, has been extended in the monograph [5] to a more general setting ofcomplete metric spaces which satisfy an appropriate geometric type of assumption.In this section we present a concise overview of these results. The earliest knownresults regarding formulating the gradient flow equations in a purely metric setting,as well as some fundamental notions presently used originate from the Italianmathematician De Giorgi 2. De Giorgi was lead to consider this problem throughhis work on a PDE in L2(Rd) where the system under consideration was restrictedto the unit sphere. Since the unit sphere in L2(Rd) is a non-convex subset theavailable theory on linear spaces could not be applied.

In order to present the basis of the theory of gradient flows on metric spaces,several facts from the ‘calculus’ on metric spaces are needed. Throughout theremainder of this section (X, d) denotes a metric space.

Absolutely continuous curves in metric spaces

A curve γ : [a, b] → X is said to be absolutely continuous of order p ∈ [1,+∞),denoted γ ∈ ACp([a, b], X), if there is a non-negative function v ∈ Lp([a, b]) suchthat for a 6 s 6 t 6 b we have that

d(γ(s), γ(t)) 6∫ t

s

v(r)dr. (1.3.1)

This definition is inspired by the classical result of analysis which asserts that afunction γ : [a, b] → R is absolutely continuous if and only if it satisfies (1.3.1)for some non-negative L1 function v. Furthermore, the metric derivative of anabsolutely continuous curve γ : [a, b]→ X is defined by

|γ|(t) := lim[a,b]3h→0

d(γ(t+ h), γ(t))h

. (1.3.2)

If γ is of class ACp([a, b], X), then the limit in (1.3.2) exists for Lebesgue a.e.t ∈ [a, b], and it equals the smallest function v ∈ Lp([a, b]) (in the Lebesgue a.esense) that satisfies (1.3.1) (see [5] Theorem 1.1.2.).

Notice that if X is a Hilbert space, and γ ∈ ACp([a, b], X) (for some a < b,a, b ∈ R, p ∈ [1,+∞)) and γ is differentiable at t ∈ [a, b], then |γ|(t) is just thenorm of the derivative γ(t) = limh→0

γ(t+h)−γ(t)h .

2The author has learned about the early development of the theory of gradient flows on metricspaces through a personal communication with Professor Luigi Ambrosio

14

Section 1.3

Metric slope of a functional

Another important concept in the context of gradient flows on metric spaces is themetric slope of a functional defined on X. As in Hilbert spaces, a functional definedon a metric space X is by definition a mapping ϕ : X → (−∞,+∞] = R∪ +∞.The metric slope denoted |∂ϕ| associated to a functional ϕ : X → (−∞,+∞], isdefined by

|∂ϕ|(x) := lim supy→x

(ϕ(x)− ϕ(y))+

d(x, y), x ∈ D(ϕ), (1.3.3)

where D(ϕ) := x ∈ X|ϕ(x) < +∞ denotes the (proper) domain of ϕ. Nowobserve that if X is a Hilbert space and ϕ differentiable at x, then |∂ϕ|(x) =| −∇ϕ(x)|. In light of (1.1.1), this observation clarifies the relevance of the metricslope of a functional: in a general metric space X, one cannot define −∇ϕ asthe geodesics are not extendable in general, but one can still define the quantity| −∇ϕ| at each x ∈ D(ϕ), i.e. ‘the length of the direction of the steepest descend’at each point in the domain of any given functional.

Metric reformulation of the Cauchy problem: the Evolution VariationalInequality (EVI)

In order to provide a theory of gradient flows in any metric space setting (whereno linear or even convex structure in the classical sense is available), one mustreformulate the abstract Cauchy problem

x(t) ∈ −∂ϕ(x(t)), L1 a.e. t > 0, x(0) = x0 ∈ D(∂ϕ)(= D(ϕ)), (1.3.4)

(L1 denotes the restriction of one-dimensional Lebesgue measure to [0,+∞)) in thefirst place. The objects that we have at hand to give any alternative formulationof the abstract Cauchy problem, are the functional ϕ, the distance function d, themetric slope |∂ϕ|, and the metric derivative of absolutely continuous curves.

In order to give an alternative purely metric formulation of the evolution prob-lem (1.3.4) in a Hilbert space H, notice that for any (x, h) ∈ D(ϕ)×H, h ∈ ∂ϕ(x)holds if and only if

〈h, z − x〉 6 ϕ(z)− ϕ(x), ∀z ∈ D(ϕ), (1.3.5)

provided that ϕ is convex. If ϕ is α-convex, i.e. if x 7→ ϕ(x) + α2 |x|

2 is convex,then h ∈ ∂ϕ(x) if and only if

〈h, z − x〉+α

2|x− z|2 + ϕ(x) 6 ϕ(z), ∀z ∈ D(ϕ). (1.3.6)

Well, suppose that ϕ : H → (−∞,+∞] is α-convex. Then in light of the charac-terisaton in (1.3.6), a curve 0 6 t 7→ x(t) ∈ X satisfies (1.3.4) if and only if foreach z ∈ D(ϕ)

〈−x(t), z − x(t)〉+α

2|x(t)− z|2 + ϕ(x(t)) 6 ϕ(z), L1-a.e. t > 0,

x(0) ∈ x0 ∈ D(ϕ).(1.3.7)

15


Observe moreover that ddt

12 |x(t) − z|2 = 〈x(t), x(t) − z〉 holds for each z ∈ H

and each t > 0 where t 7→ x(t) is differentiable ( t 7→ 12 |x(t) − z|2 is absolutely

continuous on compact subset of (0,+∞) due to the assumption on t 7→ x(t)).Hence (1.3.7) can be rewritten as follows

ddt

12d2(x(t), z)2 +

α

2d2(x(t), t) + ϕ(x(t)) 6 ϕ(z), L1-a.e. t > 0, ∀z ∈ D(ϕ),

x(0) = x0 ∈ D(ϕ),(1.3.8)

where d denotes the usual distance on H induced by its inner product. Theinequality appearing in (1.3.8) is called the Evolution Variational Inequality (EVI)in the literature (see for instance [5]). The advantage of this formulation is thatit can be posed in any metric space. Indeed, given a metric space X, a functionalϕ : X → (−∞,+∞], and a point x0 ∈ D(ϕ), one can also pose the problem of theexistence of a curve 0 6 t 7→ x(t) ∈ X, which is continuous on [0,+∞), starts atx0, and is absolutely continuous on compact subsets of (0,+∞), such that (1.3.8)holds. Nevertheless, in general existence cannot be proved. In fact it is well knownthat if X is a Banach space but not a Hilbert space, even if ϕ is convex and lowersemi-continuous, there may be no solution to this problem, while in Hilbert spacessolutions are known to exist only if ϕ is convex, or has compact lower level sets.

Curves of maximal slope

There is another characterisation of the gradient flow equations in Hilbert spaces,which is expressed in terms of the metric slope of ϕ and the metric derivative ofthe solution curve, and such formulation is also suitable to pose in general metricspaces. We will give this characterisation below, but we remark that it is not usedelsewhere in this thesis, and may be skipped.

Let a curve x : [0,+∞) → X be a solution of the Cauchy problem (1.1.1).Well, x(t) := d

dtx(t) = −∇ϕ(x(t)) for t > 0 implies that

ddtϕ(x(t)) = 〈∇ϕ(x(t)), x(t)〉 = −|∇ϕ(x(t))||x(t)|, t > 0, (1.3.9)

and

|x(t)| = |∇ϕ(x(t))| = | − ∇ϕ(x(t))|, ∀t > 0, (1.3.10)

must hold as well. Conversely, 1.3.9 implies that the vectors ∇ϕ(x(t)) and x(t) arenegative scalar multiples of each other, which together with 1.3.10 implies (1.1.1).Furthermore, (1.3.9) and (1.3.10) hold if and only if

ddtϕ(x(t)) = −1

2|x(t)|2 − 1

2| − ∇ϕ(x(t))|2, t > 0. (1.3.11)

16

Section 1.3

Indeed, (1.3.11) follows directly from (1.3.9) and (1.3.10), while (1.3.11) gives that

− |x(t)|| − ∇ϕ(x(t))|(Cauchy-Schwarz)

6 〈∇ϕ(x(t)), x(t)〉 =ddtϕ(x(t))

(1.3.11)= −1

2|x(t)|2 − 1

2| − ∇ϕ(x(t))|2

(Y oung)

6 −|x(t)|| − ∇ϕ(x(t))|, t > 0.

(1.3.12)Hence for each t > 0 the inequality may be replaced by the equality in (1.3.12),which gives (1.3.9). (1.3.10) follows as 1

2 (|x|(t)− |∇ϕ(x(t))|)2 6 0 for t > 0, thenholds too. Furthermore, as we always have that

d

dtϕ(x(t)) = 〈∇ϕ(x(t)), x(t)〉 > −| −∇ϕ(x(t))||x(t)| >

> −12| − ∇ϕ(x(t))|2 − 1

2|x(t)|2, t > 0,

(1.3.12) holds if and only if

d

dtϕ(x(t)) 6 −1

2| − ∇ϕ(x(t))|2 − 1

2|x(t)|2, t > 0. (1.3.13)

Thus another alternative formulation of the Cauchy problem reads: given ametric space (X, d) and a functional ϕ : X → (−∞,+∞], find a continuous curvex : [0,∞)→ X, which starts at x0 ∈ X of class AC2([a, b], X), such that

ddtϕ(x(t)) 6 −1

2|ϕ|2(x(t))− 1

2|x|2(t), t > 0. (1.3.14)

Solutions to (1.3.14) are called curves of maximal slope 3. We stress that noassertion regarding the existence of such curves has been made in the general case(i.e. without assumptions on X and ϕ).

In light of the classical results (in particular since Banach spaces turned out notto be a good setting for generalising gradient flow theory without further assump-tions, such as compactness of the level sets of the functional under consideration),some geometric properties of the underlying space X should clearly play the keyrole in developing the theory of gradient flows on metric spaces.

Convexity along geodesics in metric spaces

As for the notion of convexity of functionals defined on metric spaces the gener-alisation is not very hard to guess, at least if one takes a geometric point of view.If X is a Hilbert space, and ϕ : X → (−∞,+∞] is a functional, then ϕ is bydefinition convex if and only if for each pair of points x, y ∈ X one has that

ϕ((1− t)x+ ty) 6 (1− t)ϕ(x) + tϕ(y), ∀t ∈ [0, 1]. (1.3.15)

Well the curve t 7→ γ(t) = (1 − tx) + ty is a straight line, or said differently ageodesic. An equivalent characterization is given by requiring that for 0 6 s 6 t 6

3The definitions of metric slope and of curves of maximal slope are attributed to Di Giorgi.

17


1 one has that d(γ(t), γ(s)) = |t−s|d(γ(0), γ(1)), and this is the standard definitionof geodesics (or geodesic curves) in metric spaces. Hence a natural definition of theconvexity of a functional ϕ defined on a metric space X reads: for each geodesicγ : [0, 1]→ X we have that

ϕ(γ(t)) 6 (1− t)ϕ(γ(0)) + tϕ(γ(1)), ∀t ∈ [0, 1]. (1.3.16)

Furthermore, the notion of α-convexity for α ∈ R can be defined by a similar in-equality as (1.3.6) for all geodesics γ in X, but with the term α

2 t(1−t)d2(γ(0), γ(1))

added to the right hand side of the inequality.

Comparison geometry: positively and non-positively curved spaces

If one analyzes the structure of the proofs of the Hilbert space case theorems,one may observe that the following identity plays a crucial role. For any triple ofpoints v, x0, x1 ∈ X, the geodesic t 7→ x(t) := (1− t)x0 + tx1 satisfies

|v − x(t)|2 = (1− t)|v − x0|2 + t|v − x1|2 − t(1− t)|x0 − x1|2. (1.3.17)

One way to study the geometric aspects of metric spaces is to compare distancesbetween points laying on geodesic triangles with distances between points on theassociated comparison triangles. More precisely, if X is a metric space, x, y, z ∈ X,and γ1, γ2 and γ3 are geodesics joining x with y, x with z, and y with z respectively,then the associated comparison triangle is given by any triple of points x, y, z ∈ R2

such that d(x, y) = |x − y|, d(x, z) = |x − z|, d(y, z) = |y − z|. The particularchoice of such a triple of points in R2 is immaterial since any two such a triplesare isomorphic. The so called non-positively curved spaces, also known as CAT(0)spaces, present an important class of metric spaces, and these spaces are definedby the following two properties:

1. For all x0, x1 ∈ X there is a geodesic joining x0 with x1

2. For all v, x0, x1 ∈ X and for each geodesic γ joining x0 with x1 the followingholds:

d2(v, γ(t)) 6 (1− t)d2(v, x0) + td2(v, x1)− t(1− t)d2(x0, x1), ∀t ∈ [0, 1].(1.3.18)

Observe that convex subsets of a Hilbert spaces are CAT(0) spaces (see (1.3.17)).Furthermore, since R2 is a Hilbert space, the property 2. above equivalently statesthat for all v, x0, x1 ∈ X, a given geodesic γ that joins x0 with x1, and for eachassociated comparison triangle v, x0, x1 ∈ R2, denoting γ(t) := (1 − t)x0 + tx1,t ∈ [0, 1], we have that

γ(t) 6 γ(t), ∀t ∈ [0, 1]. (1.3.19)

A Riemannian manifold equipped with its Riemannian metric is a CAT(0) space ifand only if it is simply connected and has globally non-positive sectional curvature.

18

Section 1.3

Next to comparing the geometry of triangles in a metric space X with thegeometry of comparison triangles in R2, one can compare the geometry of quadri-laterals. The so called notion of the upper Alexandrov angle determined by pairsof geodesics is also an important notion in the study of the metric geometry, andvarious geometric properties of metric spaces can be expressed by comparing an-gles between geodesics. The metric spaces of arbitrary upper and lower curvaturebounds are defined by comparing the geometries of triangles in the metric spaceunder consideration with the geometries of comparison triangles in the hyperbolicspace of constant curvature equal to k < 0 and in the sphere of curvature k > 0,respectively. These ideas have been introduced by the famous Russian mathe-matician Alexandrov in 1940’s, and the theory of metric geometry, also known ascomparison geometry has become a well developed field of mathematics.

Another important class of metric spaces that has been studied extensively,are the so called positively curved spaces. A particularly important example inlight of the optimal transportation theory, gradient flows on metric spaces, andparticular in the context of the thesis, are the Wasserstein-2 spaces (P2(Rd),W2),d ∈ N, defined in Section 1.2. By definition, a metric space X is positively curvedif the inequality (1.3.18) holds with the > instead of the 6 sign.

Milestones of the theory

Knowing some basics from the theory of metric geometry is rather useful in or-der to understand some crucial elements in the theory of gradient flows in metricspaces. As we already mentioned, while there is a very elegant theory of gradi-ent flows in Hilbert spaces, there is no such thing in general Banach spaces, andjudging by this fact alone one may conclude that the linearity of the underlyingspace does not suffice to construct the solutions. The linear structure is not evennecessary. Indeed, the earliest published monograph where a gradient flow gener-ation theorem is proved in a purely metric spaces settings, is the paper [72] wherethe author mimics the celebrated Crandall-Liggett approach to construct gradientflow semigroups associated to convex lower semi-continuous functionals defined onCAT(0) spaces. It should also be mentioned that in an unpublished work [86], theauthors construct gradient flow curves associated to convex lower semi-continuousfunctionals defined on metric spaces with lower curvature bounds (in the sense ofAlexandrov), and by means of a very different technique. These authors assumethe functional to be moreover locally Lipschitz, which is too restrictive from thepoint of view of applications to PDE’s. Nevertheless, these results are quite usefulin the study of metric geometry, and from a geometrists point of view the ap-proach taken in [86] might be a more natural one. The seminal papers [54] and[85] present treatments of the Fokker-Planck equation and of the porous mediumequation as gradient flows in the Wasserstein-2 space (P2(Rd),W2) of probabil-ity measures on Rd, which is a non-linear metric space. These publications havecaused much excitement among mathematicians, and many other authors havesubsequently worked on related topics. Some of the important publications thatfollowed are [22] and [21]. A major breakthrough on the topic of gradient flows in

19


metric spaces at the abstract level as well at the level of various applications toPDE’s is considered to be the research monograph [5]. These authors reproducenatural counterparts of most of the classical theory of gradient flows in Hilbertspaces, under the so called generalized convexity assumption. The generalizedconvexity assumption states that for each triple of points v, x0, x1 ∈ X there mustexist a curve γ : [0, 1] → X such that γ(0) = x0, γ(1) = x1, and such that ϕ isconvex along this curve, i.e.

ϕ(γ(t)) 6 (1− t)ϕ(γ(0)) + tϕ(γ(1)), t ∈ [0, 1], (1.3.20)

and such that the inequality (1.3.18) holds for this γ. This technical assumptionallows one to regard the space X as a CAT(0) space in a sense, so that the crucialestimates of the variational apparatus can be obtained. The spaces where thisgeneralized convexity assumption holds are the Wasserstein-2 spaces (P2(Rd),W2),d ∈ N.

EVI: a sketch of the proof

The construction of the solutions of the evolution variational inequality (1.3.8) isperformed in an analogous way as in Hilbert spaces. Due to the assumptions, thefunctionals

D(ϕ) 3 y 7→ Φh(x, y) :=1

2hd2(x, y) + ϕ(y), h > 0, x ∈ X (1.3.21)

admit unique minimizers, which are called resolvents and typically denoted byJhx. The ‘exponential formula’

Stx := limn→∞

(J tn

)nx, x ∈ D(ϕ), (1.3.22)

defines a contraction semigroup of solutions, and this semigroup possesses pathregularity properties which are similar to the path properties of the gradient flowsemigroups on Hilbert spaces. The full treatment of this topic, as well as variousapplications regarding various PDE’s as flows in the Wasserstein-2 spaces can befound in [5]. The monograph [22] may also be consulted regarding the topic ofgradient flows on Wasserstein-2 spaces.

1.4 Research topics studied in this thesis

In the final section of this introduction, we discuss the topics and problems whichare considered in the subsequent chapters as well as the origine of the motivationto treat them, and present the author’s contributions.

In the trail of a sequence of seminars Professor Dr. Philippe Clement hasproposed to the author to prove that the product formula (see (1.1.30) and (1.1.31)for Trotter-Kato product formulas in Hilbert spaces) holds for the gradient flowon (P2(Rd),W2) associated to the Fokker-Planck equation

∂tρt = ∆xρt +∇ · (ρt∇V ), in D′((0,∞)× Rd), (1.4.1)

20

Section 1.4

where V : Rd → (−∞,+∞] is a convex lower semi-continuous functional. It hasbeen shown in [54], and elaborated in [5], that this equation can be interpretedas a gradient flow equation on the Wasserstein-2 space (P2(Rd),W2), and theassociated functional is the sum of the following two functionals:

V(µ) :=∫V dµ, µ ∈ P2(Rd), (1.4.2)

and

F(µ) :=

∫ρ log ρdx if µ Ld and µ = ρ · Ld

+∞ otherwise .(1.4.3)

Furthermore, it has been shown in [5] that both functionals V and F satisfy appro-priate lower semi-continuity, non-degeneracy, and generalised convexity conditions.Therefore these functionals on their own also generate gradient flow semigroupson (P2(Rd),W2), whose paths are solutions of the transport equation

∂tµt = ∇ · (∇V µt), in D′((0,+∞)× Rd), (1.4.4)

and the heat equation

∂tµt = ∆xµt, in D′((0,+∞)× Rd), (1.4.5)

respectively. In light of the classical results on product formulas (recall (1.1.30),(1.1.31)), it is indeed natural to consider proving that the approximation schemecan be split.

However, it is even more desirable to consider the problem of product formulasin a general setting of metric spaces. It turnes out that approximation theoryfor convex functionals can be provided in the context of CAT(0) spaces, and thistheory is the main result of this research project of the author. We present theseresults in Chapter 2.

Having completed the research on approximation theory in CAT(0 spaces, thefocus of investigation has been moved to proving the product formula for thesymmetric Fokker-Planck equation, which has been carried out successfully. Theversion of the product formula where the approximation steps are given by thesemigroups was proven, with the aid of a method where the gradient flows tech-niques are not used. Consequently, this method, jointly with some variationaltechniques, could be appplied to study the non-symmetric version of the Fokker-Planck equation as a flow on (P2(Rd),W2). The main result of this research assertsthat the semigroup of solutions of the non-symmetric Fokker-Planck equations hascontraction and path absolute continuity properties, which are similar to the prop-erties of gradient flow semigroups in (P2(Rd),W2). These results are obtained bycombining the variational techniques with the techniques of SDE’e and Markovprocesses, and we present them in Chapter 3.

The obtained results regarding the properties of the flow of solutions of thenon-symmetric Fokker-Planck equation (in addition to the known results on its

21


symmetric version) suggest a comparison with the relationship between the the-ory of gradient flows and flows induced by maximal monotone operators in Hilbertspaces. This observation propounds the folowing question: Is there a notion ofmaximal monotone operators on (P2(Rd),W2) and an associated Cauchy problemwith a flow of solutions in such a way that the theory of gradient flows in Wasser-stein spaces is generalized? And, can we recover the results on the non-symmetricFokker-Planck equation by means of the more general theory? These ideas areunfolded in Chapter IV, where the author develops an extension of the theory ofgradient flows on (P2(Rd),W2).

We conclude this introduction by giving a more detailed discussion of the prob-lems that are treated subsequenly and the obtained results.

Chapter II: Product formulas in CAT(0) spaces

In Chapter II of this thesis, we address the problem of proving the Trotter-Katoproduct formulas for gradient flows associated to convex lower semi-continuousfunctionals defined on complete CAT(0) spaces. The particular choice of CAT(0)spaces in this regard is due to the geometric properties of these spaces, whichhave turned out to suffice for proving the product formulas. In fact, the keyassumption in [5] which ensures existence of the solutions of the (EVI) (1.3.8)is the so called generalised convexity of the functional and of the square of thedistance function, an assumption modeled upon CAT(0) spaces. Since the Trotter-Kato aproximations take place in multiple directions simultaneously, it is not clearto the author whether the Ambrosio-Gigli-Savare generalised convexity may besufficient to yeld the product formula in any setting other than CAT(0) spaces, ingeneral.

Our proof roughly follows the strategy perceived in [58], but concepts andtools available in the setting of Hilbert spaces need to be replaced adequately.In particular, in the theory on Hilbert spaces, product formulas are proved withthe aid of other approximation theorems. These approximation theorems involvesemigroups associated to the following type of abstract Cauchy problem:

ddtx(t) = (F − I)(x(t)), (1.4.6)

where I denotes the identity map, and F is a given contraction. Proving productformulas in the more general case of CAT(0) spaces, where linear structure is notnecessarily available, can not be expected to be more direct than the classicalproofs are. Therefore we need to reproduce all of the classical concepts used inthe proof of product formulas in Hilbert spaces. A large portion of this chapter isdevoted to these problems.

A major difficulty occurs due to the lack of a satisfactory concept of weakconvergence in CAT(0) spaces, since weak convergence is used in the classicalproof in an essential manner. Therefore a different method must to be applied.We have used the techniques of ultra-limits, ultra-products, and ultra-extensions to

22

Section 1.4

overcome this difficulty. The developement of the theory of Chapter II correspondsto the exposition of [18] and [58] in the classical setting.

The results of this chapter are, nearly in full content, processed into an articlewhich is now an accepter paper by the author of this thesis in Advances in Calculusof Variations.

Chapter III: The non-symmetric Fokker-Planck equation asa flow in (P2(Rd),W2)

In Chapter III we return our attention to Fokker-Planck equations and we firstprove the Trotter-Kato formula. Recall that the Wasserstein-2 spaces (P2(Rd),W2)are positively curved spaces in the sense of Alexandrov (see [5] Theorem 7.3.2),and metrically isomorphic to a convex subset of a Hilbert space only when d = 1.Therefore, these spaces are not CAT(0) unless d = 1, which implies that the resultsof Chapter II can not be applied to the Fokker-Planck equations.

We prove the product formula for the Fokker-Planck equations with the aid ofthe associated stochastic differential equation and by applying the product formulafor linear semigroups on Banach spaces to the associated Kolmogorov backwardequation. The advantage of this method is that it can be applied to the non-symmetric Fokker-Planck equations as well, in order to interpret it as a flow onthe Wasserstein-2 space. Recall that the non-symmetric Fokker-Planck equationon Rd reads

∂tµt := ∆xµt +∇ · (bµt), in D′((0,+∞)× Rd), (1.4.7)

where b : Rd → Rd is a monotone mapping, i.e.

〈b(x)− b(y), x− y〉 > 0, ∀x, y ∈ Rd. (1.4.8)

Expressed differently, (1.4.7) reads∫ ∞0

∫Rd

(∂tψ(t, x)−∆xψ(t, x)− 〈∇xψ(t, x), b(x)〉

)dµt(x)dt = 0, (1.4.9)

holds for each ψ ∈ D((0,+∞) × Rd) = C∞c ((0,+∞) × Rd)—the set of smoothcompactly supported functions ψ defined on (0,+∞)× Rd. Since subdifferentialsof convex functions are maximal monotone operators, equation (1.4.1) is a specialcase of equation (1.4.7).

We emphasize that equation (1.4.7) can not be interpreted as a gradient flowequation on (P2(Rd),W2) on the following grounds. First, in analogy to theory onHilbert spaces, and the material presented in Chapter 10 of [5], one can introducethe following natural notion of cyclically monotone operators on (P2(Rd),W2). Amulti-valued operator A defined on (P2(Rd),W2), with A(µ) ⊂ L2(µ; Rd) for eachµ ∈ D(A), is cyclically monotone if and only if for each n ∈ N, µ1, ..., µn ∈ D(A),ξ1 ∈ A(µ1),...,ξn ∈ A(µn), and γj ∈ Γo(µj , µj+1) for j = 1, .., n, where µn+1 := µ1,we have that

n∑j=1

∫R2d〈ξj(y1), y2 − y1〉 dγj(y1, y2) 6 0. (1.4.10)

23


It is then straightforward to show that the Ambrosio-Gigli-Savare Frechet sub-differentials associated to convex lower semi-continuous functionals on Wasser-stein-2 spaces are cyclically monotone in this sense. In particular the Frechetsubdifferential of the relative entropy functional F is cyclically monotone. Recallthat for each µ = ρ ·Ld ∈ D(∂F) we have that ∇ρρ ∈ ∂F(µ). Observe also that dueto [5] Theorem 10.4.6, for any ball B(x, r) ⊂ Rd with center x ∈ Rd and radiusr > 0, the uniform (normalised) measure µB(x,r) = 1

Ld(B(x,r))1B(x,r) · Ld on this

ball is a member of D(∂F), while we have that its density function satisfies

∇(

1Ld(B(x,r))

1B(x,r)

)1

Ld(B(x,r))1B(x,r)

= 0 ∈ Rd, Ld-a.e.

Now, if there would be a convex functional ϕ : P2(Rd) → (−∞,+∞] which in-duces a semigroup whose paths are solutions of the non-symmetric Fokker-Planckequation (1.4.7), then its subdifferential ∂ϕ must be cyclically monotone. Hencetogether with the above arguments, choosing arbitrary n points in Rd, and con-sidering smaller and smaller balls around them, we would conclude that b itself isa cyclically monotone operator. But due to a theorem due to Rockefeller (see [17]Chapter II Theoreme 5), b should then also be a gradient of a convex function,which is a contradiction since not all maximal monotone operators on Rd are gra-dients. Therefore, the existing theory of gradient flows on Wasserstein-2 spacescan not be applied directly to the non-symmetric Fokker-Planck equation (1.4.7).

The following questions naturally arise from the above observations

1. Does the flow of the solutions of (1.4.9) induce a contraction semigroup on(P2(Rd),W2) which moreover has locally absolutely continuous paths?

2. If the answer to question 1 is affirmative, does this semigroup possess theregularising effect?

3. Suppose that the non-symmetric Fokker-Planck equation has a unique in-variant measure (which occurs if b is uniformly elliptic). Can solutions berecovered solely through the invariant measure, without knowing the drift?

An affirmative answer to question 2. above implies that even if the initialmeasure does not have a density, the solution µt at any time t > 0 is absolutelycontunuous with respect to the Lebesgue measure Ld, and that its density ρtsatisfies ρt ∈ W 1,1

loc (Rd) and ∇ρtρt∈ L2(µt; Rd), properties which solutions of the

symmetric Fokker-Planck equation and the heat equation possess as well. More-over, in that case the solution curves are locally absolutely continuous with respectto the W2 metric. Question 3 is natural since due to the gradient flow structureof the symmetric Fokker-Planck equations, its semigroup of solutions is uniquelydetermined by the invariant measure. In the last section of Chapter III we showthat in the non-symmetric case the answer to question 3 is a negative.

Finally, affirmative answers to questions 1 and 2 above imply that the Wasser-stein-2 space (P2(Rd),W2) is a natural ambient space for treating equation (1.4.7),

24

Section 1.4

and thus expand the variety of known applications of the optimal transportationtheory. Questions 1. and 2. are affirmatively answered in Chapter III.

After having completed his investigations, the author learned through privatecommunication with Giuseppe Savare, that the authors of [80] have already ob-tained the contraction property of the flow of solutions of (1.4.7). However, thecurrent exposition is distinguished from the exposition in [80] by actually provinga different product formula by entirely different means, and by proving the regu-larising effect for the non-symmetric Foker-Planck equations, a result which is notavailable elsewhere. In this manner, our results complement the research in [80].

The majority of the content of this chapter is processed into an article by theauthor of this thesis, and submitted to a journal.

Chapter IV: Maximal monotone operators on (P2(Rd),W2)

The theory of gradient flows in Hilbert spaces naturally extends to evolutionequations generated by maximal monotone operators. The main conclusions ofChapter III, i.e. the space (P2(Rd),W2) is a natural environment to study thenon-symmetric Fokker-Planck equation (1.4.7), while the semigroup of solutionsof this equation is not induced by a convex functional on (P2(Rd),W2), motivatethe following fundamental questions.

• Is it possible to provide a definition and a theory of maximal monotone oper-ators on (P2(Rd),W2), which generalizes the theory of gradient flows inducedby convex lower semi-continuous functionals?

• If we succeed in providing such a theory, can we recover our results on theFokker-Planck equations (1.4.7) obtained in Chapter III by these new means?

In light of many applications of the classical extension of the theory of gradientflows in Hilbert spaces to flows induced by maximal monotone operators, thefundamental questions posed above are not only natural but may as well lead tointeresting results regarding applications to PDE’s.

In Hilbert spaces, maximal monotone operators have various interesting prop-erties. In particular, the so called resolvent identity holds, that is, given a maximalmonotone operator A on a Hilbert space H, the resolvents Jh, h > 0 associated toA satisfy the so called resolvent identity

Jh1x = Jh2

(h2

h1x+

(1− h2

h1

)Jh1x

), 0 < h2 < h1, x ∈ H (1.4.11)

(see [17] Chapter III Section 4). Another interesting theorem states that conver-gence of Jhx as h ↓ 0 to the nearest point projection of x to the closed convexset D(A) holds. Such analogs of the classical theory have not been proven in theliterature yet (even for the special case of gradient flows on P2(Rd)), which is nev-ertheless not surprising since proofs of these results in Hilbert spaces argue fromthe point of view of the operators rather than functionals.

25


An absolute premise of the development of our new theory is that it must bean extension of the existing theory of gradient flows on Wasserstein-2 spaces in-duced by functionals that are convex along generalised geodesics. In particular,the Ambrosio-Gigli-Savare Frechet subdifferentials of lower semi-continuous func-tionals which are convex along generalised geodesics must be monotone accordingto our definition. we will call our operators to be maximal monotone operators ingeneralised sense.

A difficulty can be foreseen immediately when one considers giving a suitabledefinition of maximal monotone operators on Wasserstein-2 spaces, which ought toinduce solutions of an appropriately associated Cauchy problem. This difficulty isthe fact that (P2(Rd),W2) is a positively curved space and not a CAT(0) space. Inlight of the theory presented in [5], it seems that a notion of generalised monotonic-ity is required in order to obtain appropriate estimates, which conjointly with thegeneralised convexity of the function W 2

2 compensates for the positive curvature.No natural resolvents to be associated with equations (1.4.7) have been found.

In fact, it seems plausible that no natural resolvents as described in Chapter IVcan be associated with the equation (1.4.7). In this case the results of ChapterIII, together with [5] and Chapter IV suggest an investigation on a Wasserstein-2version of the Hille-Yosida theorem. In the context of Hilbert spaces, this theoremasserts that any contraction semigroup with continuous paths defined on a closedconvex subset of a Hilbert space is induced by a maximal monotone operator (see[17] Theorem 4.1).

A perspective regarding results of Chapter IV is a treatment of the infinitedimensional equation by a (yet to be provided) extension of our results to theinfinite dimensional setting. This program seems quite interesting, and it wouldcomplement work by other authors on gradient flows in Wasserstein spaces overinfinite dimensional spaces (see [7], and [6] for results on gradient fows in Waser-stein spaces over Hilbert spaces, see the fundamental work [41] on the Mongeproblem in the Wiener space, see [40] and [96] for treatments of the infinite di-mensional Fokker-Planck equation as a gradient flow in the Wasserstein spaceover a Cameron-Martin space by means of compactness arguments, and [70] foran extension of the previous two works by means of convexity arguments).

Chapter V: Existence of invariant measure for stochastic delay differ-entrial equations

In the final Chapter V a problem from the field of stochastic analysis is addressed.The main objective is proving the existence of an invariant measure for stochasticdelay differential equation, where the delay occurs in the drift and diffusion term.The noise is assumed to be a Levy process which does not make too many largejumps, and the diffusion coefficient is assumed to be bounded and to satisfy alocally Lipschitz condition. An essential tool used in the proof is the so calledvariation-of-constants formula for this type of equation. This formula was notpriorly available and a large part of the chapter is devoted to proving it. The maincontribution of the presented results with respect to [92] and [93] is our relaxation

26

Section 1.4

of the global Lipschitz condition to a local one. This investigation was conductedby the author in the initial period of his appointment as a PhD student at theLeiden University, and subsequently processed into an article in collaboration withOnno van Gaans. This article has been accepted for publication in in Annales del’Institut Henri Poincare, Probabilites et Statistiques.

27

Chapter 2

Approximation for convexfunctionals on non-positivelycurved spaces and theTrotter-Kato productformula

This chapter has been accepted for publication as: I. Stojkovic, “Approximation forconvex functionals on non-positively curved spaces and the Trotter-Kato productformula” in Advances in Calculus of Variations.

We study the validity of Trotter-Kato product formula in the setting of gradientflows on CAT(0) metric spaces. We follow the strategy of the proof of the Hilbertspace version of this theorem given by Kato-Masuda, but instead of the linearstructure and inner product we have only geodesics and the CAT(0) inequalityavailable. Thus we construct a counterpart of the approximation semigroups andtheir resolvents in CAT(0) spaces. We show that the convergence of the approx-imated resolvents to the resolvents of the sum functional implies convergence ofthe approximation semigroups to the semigroup associated to the sum functional.We also show that this resolvents convergence implies the product Trotter-Katoformula in CAT(0) spaces. These approximation theorems are of independentinterest. A major difficulty compared to the linear theory is the lack of an appro-priate notion of weak convergence on such spaces. This difficulty is successfullyovercome with the aid of an ultra-filters technique. The main results of our in-vestigation are various versions of the Trotter-Kato product formula in CAT(0)spaces.

29

CHAPTER 2: Approximation for convex functionals on non-positively curved spaces and

the Trotter-Kato product formula

2.1 Introduction

During the past few decades much of the classical analysis of linear spaces hasbeen extended to a metric spaces setting of one or another kind. One such strikingdevelopment is the theory of gradient flows on metric spaces. In the monograph [5],the authors L.Ambrosio, N.Gigli and G.Savare give a rather extensive treatmentof this topic adopting a purely metric point of view. In their work these authors,inspired by fundamental ideas of De Giorgi (such as metric slope and curves ofmaximal slope) and by the discoveries of Jordan-Kinderleher-Otto [54] and Otto[85], the authors of [5] construct a metric counterpart of a large part of the gradientflow theory on Hilbert spaces (which can be found in [17]). One can also consultthe work of Carrillo-McCann-Villani [22] for related topics or Villani [107], [106]for a very detailed exposition of the optimal transportation theory.

The main aim of this chapter is proving the Trotter-Kato product formula(see [58]) for gradient flows on complete CAT(0) spaces (i.e. complete geodesicmetric spaces which are non-positively curved in the sense of Alexandrov). Anintegral part of the theory of gradient flows and maximal monotone operators onHilbert spaces are various approximation theorems and in particular the productformula (see [17] chapter IV). The same holds for the theory of accreative operatorson Banach spaces, and these results are generalizations of approximation theoryfor linear C0-semigroups on Banach spaces (see Engel-Nagel [37] Chapter III).The product formula originates from Trotter’s product formula for the matrixexponentials published in 1959, which asserts that for any pair of complex matricesA,B ∈ Cd2

eA+B = limk→+∞

(eAk e

Bk )k. (2.1.1)

The importance of various approximation theorems for contraction semigroupsin different contexts is of theoretical as well as of applied character. One of the verywell known examples on the theoretical side are the appropriate approximationsof a given contraction semigroup in the proof of the Hille-Yosida theorem—whichis the key step in the proof of this famous theorem. One can consult [37] for thetheory of linear semigroups and [17] for a nonlinear counterpart. Approximationtheorems can also be used to show that certain properties are inherited by thelimit. Moreover one might encounter a situation where a product formula is theonly way to construct a semigroup corresponding to the ‘sum’ of two generators(see Chapter 3). On the applied side there are instances where one has two convexfunctionals which generate semigroups such that for each of these flows one hasnice expressions, whereas the flow induced by the sum functional is hard to handle.This occurs for instance if one wants to impose a convex constraint on an equation,i.e. force solutions to stay in a convex domain. This can be achieved by adding theindicator functional of this domain to the original one (see [58] Section Examples).In such a situation the product formula is applied to gain insight in the constrainedsolution which may be difficult to analyze directly from the variational inequalities.

It seems that the product formula has not been proved at a reasonable level

30

Section 2.1

of generality beyond the setting of linear spaces—therefore we consider our effortsappropriate. The product formula in the linear setting is a consequence of severalother approximation theorems. In particular a theorem stating that convergence ofresolvents implies convergence of semigroups, and these results are of independentinterest (see [18] for the theory on Banach spaces). Dropping the assumption oflinearity of the underlying space does not allow for straightforward generalisationsand requires new ideas.

CAT(0) spaces have been attracting much attention already since the birth ofcomparison geometry in 1940’s. Being the natural generalization of non-positivelycurved Riemannian manifolds, CAT(0) spaces have very interesting geometricproperties. As a matter a fact the larger class of metric spaces with one-sidedcurvature bounds in the sense of Alexandrov appear to be the most studied non-smooth geometric objects up to the current date. The foundations of this theoryare now well developed and a detailed exposition of this topic can be found in [19],[20] and [4] among other monographs.

However not only purely geometric aspects of CAT(0) spaces have been studiedup to now. We will devote some lines to giving some references about published pa-pers where different authors considered various generalisations of theories on linearspaces in the CAT(0) setting. We stress however that the presented informationis not necessary in order to follow our exposition in the subsequent sections.

The theory of harmonic maps on Riemannian manifolds has been extendedto CAT(0) spaces during the 1990’s by various authors. This theory initiatedby M.Gromov and R.Schoen in light of some applications to the group theory(see [46]). This work was followed by a series of remarkable papers by R.Schoen,N.Korevaar and J.Jost (see [60], [55], [56] and [57]), and many more by differentauthors.

There have also been many instances where various authors studied monotoneand accreative operator theory and the corresponding semigroups in a non-smoothsetting. In particular on a Hilbert ball, and among other works we have mono-graphs [2], [89], [90], [91] by S.Reich etal. Moreover these issues in a Hilbertmanifold have been the subject of investigations (see [50]). See also [51] for atreatment of a ‘wave-like’ equation on a Hilbert manifold by Iwamiya and Okochi.These authors make no assumptions about the curvature—instead their assump-tions on the functional are rather heavy. Recently in [64] and [65] Li, Lopez andMartin-Marques treated a problem involving monotone vector fields on Hadamardmanifolds in relation with some optimization problems (see also [88]).

A gradient flow generation theorem in CAT(0) spaces was provided by U.Mayerin [72] and this is the first published result on gradient flows beyond a setting oflinear spaces. However there is also the unpublished work of G.Perelman andA.Petrunin [86]. K.T.Sturm developed a large portion of stochastics for CAT(0)valued stochastic processes, such as Markov operators and martingales (see [101],[25], [99] and [100]). Moreover in 2007 Larotonda [62] showed that the set ofpositive invertible unitized Hilbert-Schmidt operators on a Hilbert manifold is acomplete CAT(0) space (see also references in [62] for many more papers about re-lated topics, such as [42]). Finally let us mention works of S-I.Ohta [82] and [83] in

31



which he introduces Sobolev spaces of functions with CAT(0) targets and considersflows on the Wasserstein spaces over compact CAT(0) spaces, respectively.

To conclude this introduction, let us give the content of the remainder of thechapter. Section 2 contains some background material. In Section 3 we constructthe approximation semigroups which we need for proving the Trotter-Kato productformula. We also show two convergence theorems. In particular a theorem whichasserts that convergence of resolvents implies convergence of the correspondingsemigroups. We will not formulate nor prove these theorems at the same levelof generality as it has been done in the linear setting for one should first give atreatment of maximal monotone operators on CAT(0) spaces. However this ia awork in progress by the author and his collaborator. This section correspondsto the monograph [18] by Brezis-Pazy and the treatment of our Lemma 2.3.11on Banach spaces can be found in [73]. In Section 4 we prove the Trotter-Katoproduct formula on CAT(0) spaces and we follow the strategy of Kato-Masuda in[58] as far as possible in our setting. However in [58] the authors use the weakconvergence on Hilbert spaces, and there is no suitable counterpart of it on generalCAT(0) spaces (but see [34] and [59] for some results in this direction). We willcircumvent the lack of a suitable notion of weak convergence by using ultra-filters.This idea is inspired by successful employment of such a technique by Lytchak in[69].1 Contrary to Lytchak’s work, our functionals are only supposed to be lowersemi-continuous and not locally Lipschitz.

2.2 Preliminaries

In this section we present the basic concepts and facts that we need in the remain-der of the exposition.

Definition 2.2.1. Let (X, d) be a metric space. A curve x : [a, b] → X is calleda geodesic if for all s, t ∈ [a, b]

d(x(s), x(t)) = |t− s|d(x(a), x(b)). (2.2.1)

Such a curve is said to connect x0 with x1 if x(a) = x0 and x(b) = x1 and we willmoreover say that x emanates from x0 in such case. X is a (uniquely) geodesicspace if any pair of points x0 and x1 can be connected by a (unique) geodesic. Bycomposing with an affine increasing homeomorphism we can always assume ourgeodesics to be defined on [0, 1]. A subset A ⊂ X is (geodesically) convex if theimage of each geodesic that connects pairs of points x 6= y ∈ X is contained in A.

Definition 2.2.2. A geodesic metric space X is called CAT(0) space 2 if for anypair of points x0, x1 ∈ X and a (base) point y ∈ X each geodesic x : [0, 1] → X

1During the author’s visit to Munster, Germany in August 2010, his host Anton Petrunintold the author about the work [69]. Author wishes to thank Petrunin for his kind and usefuladvice

2Some authors call such spaces non-positively curved spaces.

32

Section 2.2

connecting x0 and x1 satisfies the following (CAT(0)) inequality:

d2(y, x(t)) 6 (1− t)d2(y, x0) + td2(y, x1)− t(1− t)d2(x0, x1) (2.2.2)

for all t ∈ [0, 1].

It is easy to show that CAT(0) spaces are uniquely geodesic.

Proposition 2.2.3. If X is a CAT(0) space then it is uniquely geodesic.

Proof. See [19] Chapter II Proposition 1.4 (1).

In the sequel we will use the following notation. If X is a CAT(0) space,x, y ∈ X and t ∈ [0, 1], then

(1− t)x⊕ ty (2.2.3)

denotes the point γ(t) on the unique geodesic γ : [0, 1] 7→ X which connects x andy.

Example 2.2.4. 1. A convex subset of a Hilbert space is a CAT(0) space.Moreover a normed space is CAT(0) only if its norm is generated by aninner product.

2. A C3 simply connected Riemannian manifold with non-positive sectionalcurvature equipped with its Riemannian metric is a CAT(0) space.

3. The set of positive unitized Hilbert-Schmidt operators on a Hilbert space canbe given a Riemannian structure that makes it a smooth Hilbert manifoldwhich is metrically complete, geodesic and globally non-positively curved,i.e. a complete CAT(0) space. See [62] and references for more examples ofnon-positively curved Hilbert manifolds that are defined as a certain subsetof the operator space of a Hilbert space.

4. Euclidean Bruhat Tits buildings are CAT(0) spaces.

5. The unit ball in a Hilbert space equipped with the hyperbolic metric is aCAT(0) space (see [45]).

6. Let X be a CAT(0) space, let (M,M,m) be a measure space and considerthe set Ω of L2 maps f : M → X (i.e. for some hence all points x ∈ X∫Md2(f(ω), x)dm(ω) < +∞), and define the distance on Ω by

d2(f1, f2) :=∫M

d2(f1(ω), f2(ω))dm(ω).

Then (Ω, d) is a complete CAT(0) space.

7. The spaces W 1,2Φ (Ω, X) introduced in [72] 3.2 Section 6 are CAT(0) spaces

as is shown there.

33



There are several equivalent characterizations of CAT(0) spaces. In order toformulate these we need to introduce two concepts which are basic tools in thestudy of geometry in a non-smooth setting.

The first one is the concept of a comparison triangle which was extensivelyused by the Russian mathematician Alexandrov and his school in 1940’s, thoughit seems to appear for the first time in the work of the Australian mathematicianWald. Given any triple of distinct points p, q, r in a metric space X there is aunique up to isometry triangle in R2 with vertices p, q, r ∈ R2 such that

d(p, q) = |p− q|, d(q, r) = |q − r|, d(p, r) = |p− r|, (2.2.4)

where |.| denotes the Euclidean norm on R2. This follows by the triangle inequalityfor the distance d. Given any pair of points q,r in R2 the line segment [q, r] is bydefinition the set of convex combinations of points q and r in the linear space R2,i.e.

[q, r] = (1− t)q + tr t ∈ [0, 1].

More generally, given a pair of points q and r in our metric space X and a geodesicγ : [0, a] → X with end points γ(0) = q and γ(1) = r the geodesic segment [q, r]is the image of γ. If X is uniquely geodesic, then [q, r] is independent of theparticular choice of a geodesic γ.

If p, q, r are distinct points in X and γ1, γ2, γ3 is a choice of geodesics joiningthem, then the corresponding geodesic triangle denoted by ∆ = ∆(p, q, r) is theset of points in the union of the images of these three geodesics. The correspondingcomparison triangle is usually denoted by ∆ = ∆(p, q, r). Note that the geometryof ∆ depends only on the quantities d(p, q), d(p, r) and d(q, r), not on γ1, γ2, γ3.

For any triple of points p, q, r ∈ R2 (or in any Hilbert space), and for eacht ∈ [0, 1] the point x = (1− t)q + tr ∈ [q, r] satisfies the following identity:

|p− x|2 = |p− ((1− t)q+ tr)|2 = (1− t)2|p− q|2 + t2|p− r|2 + t(1− t)|p− q||p− r|.

Now we can easily see that a geodesic metric space X is CAT(0) if and onlyif for any triple of distinct points p, q, r ∈ X any choice of geodesic segment [q, r]and any point x = x(t) ∈ [q, r] the (unique up to isometry) comparison triangle∆ = ∆(p, q, r) for the geodesic triangle ∆ = ∆(p, q, r) satisfies

d(p, x) 6 |p− x(t)| (2.2.5)

where x(t) := (1− t)q + tr.The second concept is the Alexandrov upper angle between geodesics. In a

metric space geodesics could be thought of as straight lines. In a flat space (likeRn or more generally a convex subset of a Hilbert space) if p, q, r are points theangle ∠p(q, r) between line segments [p, q] and [p, r] (the geodesics are unique inthis situation) is given by the cosine law

|q − r|2 = |q − p|2 + |p− r|2 − 2|q − p||p− r| cos ∠p(q, r). (2.2.6)

34

Section 2.2

Equivalently

cos ∠(~x, ~y) =〈~x, ~y〉|~x||~y|

(2.2.7)

for any pair of vectors ~x and ~y in the flat space (and we adopt the convention thatangles assume values in [0, π]).

If the space under consideration is a Riemannian manifold M , than the anglebetween two geodesic segments with common end point is naturally defined asangle between their velocity vectors at this end point. Both of these vectors areelements of the tangent space TpM to M at p, and the angle between them iscomputed as in (2.2.6). We can express this quantity purely in terms of thedistance function, and subsequently use that expression as the definition of the(upper or lower) angle between geodesics in any metric space.

Definition 2.2.5. Let p be a point in a metric space X, and let γ1, γ2 : [0, 1]→ Xbe two geodesics emanating from p, i.e. γ1(0) = γ2(0) = p. For each s, t ∈(0, 1] let ∆s,t := ∆(p, γ1(s), γ2(t)) denote a comparison triangle for the triangle∆(p, γ1(s), γ2(t)) ⊂ X. Let moreover αs,t be the angle in ∆s,t at vertex p. TheAlexandrov upper angle between γ1 and γ2 is defined by

∠p(γ1, γ2) := lim sups,t↓0

αs,t (2.2.8)

Equivalently one can define

∠p(γ1, γ2) := limε→0

sups∈(0,ε);t∈[0,1]

αs,t (2.2.9)

(for proof see [19], Part I Proposition 1.16). Notice that this definition coincideswith (2.2.7) if X = Rn, and also that ∠p(γ1, γ2) is in fact defined in terms of thedistance function via (2.2.6).

We remark that if p, q, r ∈ X and γ1, γ2 are geodesics with γ1(0) = γ2(0) = p,γ1(1) = q, γ2(1) = r, we write ∠p(q, r) instead of ∠p(γ1, γ2) or ∠([p, q], [p, r])provided there is no ambiguity.

The notion of angle gives rise to two other characterizations of CAT(0) spaces,and we state all of these characterizations in the following proposition.

Proposition 2.2.6. Let X be a geodesic space. Then the following conditions areequivalent

1. X is a CAT(0) space

2. For every geodesic triangle ∆(p, q, r) in X and every point x ∈ [q, r], thecomparison point x ∈ [q, r] ⊂ ∆(p, q, r) ⊂ R2 satisfies

d(p, x) 6 |p− x| (2.2.10)

35



3. For every geodesic triangle ∆(p, q, r) in X and every pair of points x ∈ [p, q],y ∈ [p, r] distinct from p, the angles at the vertices corresponding to p in thecomparison triangles ∆(p, q, r), ∆(p, x, y) ⊂ R2 satisfy

∠p(x, y) 6 ∠p(q, r) (2.2.11)

4. The Alexandrov upper angle between two sides of a geodesic triangle in X isnot greater than the angle between the corresponding sides of its comparisontriangle in R2.

5. For every geodesic triangle ∆([p, q], [p, r], [q, r]) in X with p 6= q and q 6= r,if γ denotes the Alexandrov upper angle between [p, q] and [p, r] at p and if∆(p, q, r) ⊂ R2 is a geodesic triangle with |p− q)| = d(p, q), |p− r| = d(p, r)and ∠p(q, r) = γ, than d(q, r) > |q − r|

If any of the above conditions holds, then for each triple of points p, q, r ∈ Xwe have

d2(q, r) > d2(p, q) + d2(p, r)− 2d(p, q)d(p, r) cos ∠p(q, r). (2.2.12)

Proof. For equivalence between items 1,2,3,4 and 5, see [19] Part II Proposition1.7. For inequality (2.2.12) see [19] Part II Exercise 1.9.

Remark 2.2.7. In exactly the same way one can compare the geometry of trian-gles in a metric space X with other model spaces to define CAT(k) spaces for allk ∈ R, by requiring angles to be thinner than in the model space. For k = 1 themodel space is the unit sphere S2 equipped with the angular metric (see [19] PartI Proposition 2.1) given by

cos d(A,B) := 〈A,B〉, A,B ∈ S2, (2.2.13)

where by definition d(A,B) ∈ [0, π]. For other k > 0 this distance is multiplied by1k .

For k = −1 the model space is the hyperbolic space H2, and one of the possibleways to define it is the following. Consider the symmetric bilinear form on R3 givenby 〈x|y〉 := x1y1 +x2y2−x3y3. Then H2 = x ∈ R3|〈x|y〉 = −1, x3 > 0, equippedwith the distance function given by

cosh d(x, y) := −〈x|y〉. (2.2.14)

For other k < 0 one rescales this distance by 1√−k .

One may consult [19] for a thorough treatment of various aspects of the theoryof CAT(k) spaces.

However in this thesis we restrict our consideration to CAT(0) spaces. Thereis also a characterization of CAT(k) spaces based on comparison of quadriliterals(see [19] Part II Proposition 1.11).

36

Section 2.2

Proposition 2.2.8. If X is a CAT(0) space then the distance function d : X ×X → R>0 is convex, i.e. given any pair of geodesics γ1, γ2 : [0, 1] → X thefollowing inequality holds

d(γ1(t), γ2(t)) 6 (1− t)d(γ1(0), γ2(0)) + td(γ1(1), γ2(1)), ∀ t ∈ [0, 1]. (2.2.15)

Moreover for each point y ∈ X and any geodesic γ : [0, 1] → X the followinganalogue of the triangle inequality holds

d(y, γ(t)) 6 (1− t)d(y, γ(0)) + td(y, γ(1)), ∀ t ∈ [0, 1]. (2.2.16)

Proof. For the proof of (2.2.15) see [19] Chapter II Proposition 2.2. To showinequality (2.2.16) consider a comparison triangle ∆(p, q, r) ⊂ R2 for the geodesictriangle ∆(p, q, r) ⊂ X, and denote γ(t) := (1− t)q+ tr for t ∈ [0, 1]. Now in lightof the item 2 of Proposition 2.2.6 and the triangle inequality in R2 we estimate

d(p, γ(t)) 6 |p− γ(t)| 6 (1− t)|p− q|+ t|p− r| = (1− t)d(p, q) + td(p, r). (2.2.17)

The convexity property stated above is clearly weaker than the CAT(0) in-equality (2.2.2). Indeed the same inequality holds in uniformly convex Banachspaces (which are uniquely geodesic, and the geodesics are precisely the convexcombinations of pairs of distinct points), but as we already remarked in Example2.2.4 Hilbert spaces are the only Banach spaces that are also CAT(0).

CAT(0) spaces share another interesting property with Hilbert spaces. Namely,one can define the nearest-point projection onto a closed convex set.

Definition 2.2.9. Let X be a geodesic space. A subset A ⊂ X is convex if foreach pair of points x 6= y in A for some geodesic segment [x, y] we have [x, y] ⊂ A.

Proposition 2.2.10. Let X be a CAT(0) space and let A ⊂ X be a convex subsetwhich is complete in the induced metric. Then

1. For every x ∈ X there exists a unique point π(x) ∈ A such that d(x, π(x)) =d(x,A) := infy∈A d(x, y).

2. If x′ belongs to the geodesic segment [x, π(x)] then π(x) = π(x′).

3. Given x 6∈ A and y ∈ A, if y 6= π(x) then ∠π(x)(x, y) > π/2.

4. The map x 7→ π(x) is a retraction defined on X onto A which does notincrease distances; d(π(x), π(y)) 6 d(x, y) for x, y ∈ X and the map H :X × [0, 1] → X associating to (x, t) the point a distance td(x, π(x)) from xon the geodesic segment [x, π(x)] is a continuous homotopy from the identitymap of X to π.

Proof. See [19] Part II Proposition 2.4.

37



Remark 2.2.11. By the cosine law (2.2.12), item 3. above implies that for v ∈ A

d2(x, v)− d2(π(x), v) > d2(x, π(x)) (2.2.18)

In order to state one of our results we will also use the following general factabout metric spaces. The definition of length spaces can be found in [19] Part IDefinition 3.1; all geodesic spaces are also length spaces. As usually a topologicalspace is said to be locally compact if every point in it has a relatively compactneighborhood.

Theorem 2.2.12. (The Hopf-Rinow-theorem) Let X be a length space. If X iscomplete and locally compact then

1. Every closed bounded set is compact

2. X is a geodesic space.

Proof. See [19] Chapter I Proposition 3.7.

Next we give some facts from the theory of gradient flows on metric spaces.A map ϕ : X → (−∞,+∞] is called a functional, and it is said to be geodesi-

cally convex (and we will abbreviate to convex) if for any pair of points x0 6= x1

in X any geodesic γ : [0, 1]→ X with γ(0) = x0 and γ(1) = x1 satisfies

ϕ(γ(t)) 6 (1− t)ϕ(γ(0)) + tϕ(γ(1)). (2.2.19)

D(ϕ) := x ∈ X| ϕ(x) < +∞ is called the (proper) domain of ϕ, and ϕ issaid to be proper if D(ϕ) 6= ∅. ϕ is lower semi-continuous (shortly denoted l.s.c.)if for each sequence (xn)n ⊂ X and x ∈ X we have that

limnxn = x =⇒ ϕ(x) 6 lim inf

nϕ(xn). (2.2.20)

The sets La := x ∈ X| ϕ(x) 6 a for a ∈ R are called sub-level sets.A curve u : [a, b] 7→ X is absolutely continuous of order p > 1 denoted u ∈

ACp([a, b];X), if there is a nonnegative function m ∈ Lp([a, b]; R) such that fora 6 c 6 d 6 b

d(u(c), u(d)) 6∫ d

c

m(s)ds. (2.2.21)

If the above holds for a curve u then its metric derivative

|u|(t) := limh→0

d(u(t+ h), u(t))h

(2.2.22)

is defined L1a.e. on (a, b) and it is the smallest function m such that (2.2.21) holds(Part I Section 1 in [5] for details). Moreover a curve u : (a, b) 7→ X is locallyabsolutely continuous of order p > 1 denoted u ∈ ACploc((a, b);X), if it is absolutelycontinuous of the same order on each compact subinterval of (a, b). If (a, b) 3 t 7→f(t) is a real valued function which is moreover right differentiable at t0 ∈ (a, b)

38

Section 2.2

then we will denote ddt+

f(t0) := limh↓0f(t0+h)−f(t0)

h to be its right derivative att0. Following this notation if u ∈ ACp((a, b);X) for some p ∈ [1,+∞) we will alsodenote |u|+(t0) := limh↓0

d(u(t0+h),u(t0))h to be the right metric derivative at t0 ∈ I

if this limit exists.As it is explained in [5] there is an appropriate metric formulation of the gra-

dient flow inclusion ddtX(t) ∈ −∂ϕ(X(t)), X(0) = x ∈ D(ϕ) associated with a

convex l.s.c. functional ϕ on a Hilbert space H. ∂ϕ denotes the subdifferential of ϕ.This formulation is the so-called Evolution Variational Inequality (shortly denotedEVI), i.e. one seeks a continuous curve u : [0,+∞) 7→ X of class AC2

loc([0,+∞);X)such that for each v ∈ D(ϕ)

d

dtd2(x(t), v) + ϕ(x(t)) 6 ϕ(x) for L1 − a.e. t ∈ [0,+∞). (2.2.23)

In [5] the authors work with a certain generalized convexity assumption (seeAssumption 4.0.1 and also a weaker Assumption 2.4.5 there), which is easily seen tohold ifX is a CAT(0) space and ϕ is geodesically convex3. Next to this assumption,in [5] the functional under consideration is assumed to be bounded from below onsome ball in the space (see (2.1.2b) there). However this always holds in theCAT(0) setting. This fact is proven in [72] Lemma 1.3. Hence all of the theorypresented in [5] applies if the following assumption holds:

Assumption 2.2.13. X is a complete CAT(0) space and ϕ : X → (−∞,+∞] aproper, geodesically convex and lower semi-continuous functional.

Proposition 2.2.14. Let Assumption 2.2.13 hold. Then for any x ∈ X there areconstants c, b ∈ R such that

ϕ(y) > c+ bd(x, y) ∀y ∈ X. (2.2.24)

Proof. See [26] Lemma 4.1.

In order to construct the semigroup of solutions of the EVI and prove thatit has the contraction property, we define the resolvents of ϕ. For x, y ∈ X andh > 0 we consider functionals

Φ(h, x, y) :=1

2hd2(x, y) + ϕ(y) (2.2.25)

and we look for a minimizer y ∈ X for x ∈ X and h > 0 fixed, i.e. we look for

Jhx := argmin

12hd2(x, y) + ϕ(y) | y ∈ X

. (2.2.26)

3As a matter a fact these assumptions are modeled upon such a situation for the reason thatthe variational apparatus can be built then.

39



This ‘elliptic’ problem has a unique solution, provided the above discussed as-sumptions hold, and moreover it has the following properties 4:

Proposition 2.2.15. Grant Assumption 2.2.13. Then for each h > 0 and x ∈ Xthe functional y 7→ Φ(h, x, y) given by (2.2.25) admits a unique minimizer, denotedby Jhx. Moreover Jh : D(ϕ)→ D(ϕ) is a contraction for each h > 0.

Proof. As we already pointed out our Assumption 2.2.13 ensures the assumptionsin [5], and existence and uniqueness of minimizers are proven in Theorem 4.1.2(i) there. The contraction property of the resolvents Jh : D(ϕ) → D(ϕ) underAssumption 2.2.13 is shown in [72] Lemma 1.12.

For x ∈ X and h > 0 we denote

ϕh(x) := infy∈X

Φ(t, x, y) =1

2hd2(x,Jhx) + ϕ(Jhx) (2.2.27)

The following facts are crucial in constructing the solutions of the EVI.

Proposition 2.2.16. Grant Assumption 2.2.13. If x ∈ D(ϕ) and if xn is asequence in D(ϕ) satisfying

lim supn→+∞

Φ(t, x, xn) 6 ϕh(x) (2.2.28)

then xn → Jhx as n→ +∞.Moreover we have for v ∈ D(ϕ)

12hd2(Jhx, v)− 1

2hd2(x, v) +

12hd2(x,Jhx) + ϕ(Jhx) 6 ϕ(v). (2.2.29)

Remark 2.2.17. Notice that inequality (2.2.29) implies the defining inequalityfor Jhx, i.e. (2.2.26), and therefore seems to be stronger.

Proof. For the first statement see [5] Lemma 4.1.1. and for the second one see [5]Theorem 4.1.2 (ii).

If the metric space under consideration has one-sided curvature bounds in thesense of Alexandrov, then one can define the Euclidean tangent cones over thespace of directions at each point in space, and also define an appropriate gradientof a convex functional (see [69], [82] and [4]). We will not use these notions, howeverwe do need to use the notion of the ‘norm’ of the ‘negative of the gradient’, whichis called the metric slope and is defined as follows.5

4We have formulated the main results about the gradient flows in the setting of our As-sumption 2.2.13. We remark though that except for the contraction property of the resolvents,all the other properties hold under the generalized convexity and non-degeneracy assumptionsmentioned above. Resolvents are then still continuous in x.

5This definition has been introduced by De Giorgi.

40

Section 2.2

Definition 2.2.18. Let X be a metric space and let ϕ be a functional on X.Then for each x ∈ D(ϕ) the metric slope |∂ϕ|(x) of ϕ at x is given by

|∂ϕ|(x) := lim supy→x

(ϕ(x)− ϕ(y))+

d(x, y). (2.2.30)

If Assumption 2.2.13 holds then we may substitute lim sup by sup in (2.2.30)(see [5] Theorem 2.4.9). We may have that |∂ϕ|(x) = +∞ for x ∈ D(ϕ). Noticethat if X is a Hilbert space and ϕ is convex and C1, we have that |∂ϕ|(x) =| − ∇ϕ|(x), which clarifies the relevance of this definition.

Proposition 2.2.19. Let X be a metric space and let a functional ϕ on X begiven. Grant Assumption 2.2.13. Then for x ∈ X and h > 0 we have Jhx ∈D(|∂ϕ|) and

|∂ϕ|(Jhx) 6d(Jhx, x)

h. (2.2.31)

In particular D(|∂ϕ|) = D(ϕ) ⊂ X.If x ∈ D(|∂ϕ|) we also have the following estimates

|∂ϕ|2(Jhx) 6d2(Jhx, x)

h6 2

ϕ(x)− ϕh(x)h

6 |∂ϕ|2(x). (2.2.32)

Proof. For the first statement see [5] Lemma 3.1.3. For the second statement, see[5] Theorem 3.1.6 (2).

Finally we state the theorem about the generation of the flow and its propertiesin our setting (but we remark that exactly the same theorem is proven in [5] undergeneralized convexity and non-degeneracy assumptions).

Theorem 2.2.20. Grant Assumption 2.2.13. Let x0 ∈ D(ϕ). Then we have thefollowing:

1. (Convergence and exponential formula) For each t > 0 the limit

limn→∞

(J tn

)nx =: Stx0 (2.2.33)

exists. Moreover (St)t>0 is a semigroup on D(ϕ).

2. (Regularizing effect) x(t) := Stx0 is a curve of class AC2loc((0,+∞);X) with

x(t) ∈ D(|∂ϕ|) ⊂ D(ϕ) for each t > 0, i.e. for each ε > 0 and T > 0t 7→ x(t) is Lipschitz on [ε, T ] (actually on [ε,+∞)) and

ϕ(x(t)) 6 ϕt(x0) 6 ϕ(v) +12td2(v, x0) ∀v ∈ D(ϕ),

|∂ϕ|2(x(t)) 6 |∂ϕ|2(v) +1t2d2(v, x0) ∀v ∈ D(|∂ϕ|)

(2.2.34)

41



3. (Uniqueness and EVI) x(t) is unique solution of the EVI

12

ddtd2(x(t), v) + ϕ(x(t)) 6 ϕ(v) L1 -a.e. t > 0, ∀v ∈ D(ϕ). (2.2.35)

among all the locally absolutely continuous curves on [0,+∞) such thatlimt↓0 x(t) = x0.

4. (Contraction semigroup) The map (x, t) 7→ St(x) : D(ϕ)× [0,+∞)→ D(ϕ)(where by definition S0(x) := x ∀x ∈ D(ϕ)) is a contraction semigroup,

d(Stx, Sty) 6 d(x, y) ∀t > 0, ∀x, y ∈ D(ϕ) (2.2.36)

5. (Optimal Error estimate) If x0 ∈ D(ϕ) then

d2(Stx0, (Jt/n)nx0) 6t

n(ϕ(x0)− ϕt/n(x0)) 6

t2

2n2|∂ϕ|2(x0) (2.2.37)

for t > 0 and n ∈ N.

6. The equation

ddt+

ϕ(x(t)) = −|∂ϕ|2(x(t)) = −|x|2+(t) = −|∂ϕ|(x(t))|x|+(t) (2.2.38)

is satisfied for every t > 0

Proof. See [5] Theorem 4.0.4 for the first five items, and [5] Theorem 2.4.15 forthe last item.

Remark 2.2.21. In the setting of Theorem 2.2.20 the function t 7→ ϕ(x(t)) isnon-increasing since by (2.2.38) we have d

dtϕ(x(t)) = −|∂ϕ|2(x(t)) and t 7→ x(t)is continuous on [0,+∞), hence integrating (2.2.35) we obtain for any x0 ∈ D(ϕ)and t > 0 and v ∈ D(ϕ)

12td2(x(t), v)− 1

2td2(x(0), v) + ϕ(x(t)) 6 ϕ(v). (2.2.39)

Consider again the setting where X is a CAT(0) space and ϕ is geodesically convex.Then D(ϕ) is a closed convex set in X, and denoting π : X → D(ϕ) to be thenearest point projection defined in Proposition 2.2.10, the Remark below thatproposition gives that for any y ∈ X, t > 0 and v ∈ D(ϕ)

12td2(x(t), v)− 1

2td2(y, v) + ϕ(x(t)) 6 ϕ(v) (2.2.40)

where x(t) is the unique solution of the EVI subject to initial condition x0 := π(y).

42

Section 2.2

Next we state the classical Trotter-Kato product formula for convex function-als on Hilbert spaces. We need some notation first. H is the Hilbert space underconsideration and ϕ1, ..., ϕn are proper, convex, lower semi-continuous function-als on H. Denote Dj := D(ϕj), Ej := Dj , ϕ :=

∑nj=1 ϕj , D := D(ϕ) (6= by

assumption), E := D. Furthermore Pj denotes the nearest point projection onto

Ej ,(Sjt

)t>0

, j = 1, n denote the semigroups of solutions associated to the ϕj ’s

and(J jh)h>0

denote the resolvents associated to the ϕj ’s. Finally for t > 0 we

denoteU jt : X → X, U jt = J jt , for j = 1, ...n, t > 0, (2.2.41)

orU jt : X → X, U jt = Sjt Pj , for j = 1, ...n, t > 0 (2.2.42)

Furthermore we denote

Ft := Unt Un−1t ... U1

t : X → X, t > 0. (2.2.43)

The operators U jt defined in (2.2.43) are neede to forormulate the Trotter-Katoproduct formulas. It may occur that for some i ∈ 1, ...n the sets Ei and Ei+1 donot coincide, and since the domains of the semigroups (Sit)t> and (Si+1

t )t> equalEi and Ei+1, respectively, the mapping Si+1

t Sit may not be defined—the reasonwhy the projections Pj are used in (2.2.42).

Theorem 2.2.22. (Trotter product formula for convex functionals on Hilbertspaces) Let H be a Hilbert space, and let ϕ1, ..., ϕn be proper convex and lowersemi-continuous functionals such that ϕ :=

∑nj=1 ϕj 6≡ +∞. Then

limt↓0

[1 +

λ

t(1− Ft)

]−1

x→ Jλx for x ∈ H,λ > 0, (2.2.44)

andlimn→∞

(F tn

)nx→ Stx for x ∈ E, t > 0. (2.2.45)

Proof. (2.2.44) implies (2.2.45) and this fact is proved in [18]. (2.2.44) is provedin [58].

Essentially, the Trotter-Kato product formula assers that if the functional un-der consideration is a sum of a finite sequance of convex functionals, then a descreteapproximation scheme (see Subsection 1.1.1 for the description of the approxima-tion scheme in case of one functional) can be split, by applying the semigroupscomposed with projections Pj or the resolvents separately

The remainder of this section is devoted to ultrafilters, ultra-products, ultra-limits, and ultra-extensions. Most of this material is taken from [19] and [4].However we have modified some parts of it since we are considering lower semi-continuous functionals and not locally Lipschitz ones.

43



Definition 2.2.23. A non-principal ultra-filter on N is a finitely additive proba-bility measure ω such that each subset S ⊂ N is ω-measurable, ω(S) ∈ 0, 1 andω(S) = 0 if S is finite.

It possible to construct a non-principal ultra-filter (see exercise 5.48 in ChapterI of [19]). However, there are many distinct such objects, and none of them is acanonical one. Let us choose one such non-principal ultra-filter ω throughout theremainder of this chapter. We emphasize though that the remaining theorems andlemmas in this section will be stated in termes of arbitrary non-principal ultra-filters, since the stated claims hold foe all non-principal ultra-filter—not just forthe one that we chose above.

Definition 2.2.24. Let (an)n be a sequence in [−∞,+∞], and let ω be a no-principal ultra-filter. We say that a ∈ [−∞,+∞] is the ω-limit of (an)n and wewrite limn→ω an = a if for each neighborhood N of a in [−∞,+∞] we have thatfor ω-a.e. n ∈ N, an is in N .

Lemma 2.2.25. Let ω be a no-principal ultra-filter. For any sequence (ak)k ⊂[−∞,+∞] there is a unique l ∈ [−∞,+∞] such that l = limk→ω ak, i.e. for eachε > 0 for ω-a.e. n ∈ N we have |ak − l| < ε if l ∈ R, and ak < − 1

ε or ak > 1ε

if l = −∞ or l = +∞, respectively. We write then limk→ω ak = l and say that lis the ω-limit of ak. Moreover if limk→ω ak = l1, limk→ω bk = l2 and l1 ∈ R orl2 ∈ R then limk→ω(ak + bk) = l1 + l2 (where c +∞ = +∞ and −∞ + c = −∞for c ∈ R).

Proof. For uniqueness suppose l1 < l2 both satisfy l1 = limk→ω ak, l2 = limk→ω ak.We only treat the case l1, l2 ∈ R the other cases being analogous. Choose ε > 0such that l1 + ε < l2 − ε. Then for j = 1, 2 the sets Sj := k ∈ N| |ak − lj | < εsatisfy ω(Sj) = 1 but S1 ∩ S2 = ∅, a contradiction.

For existence define l := sups ∈ R| ∀ε > 0 for ω − a.e. k ∈ N ak > s − ε.Then for any ε > 0 ω-a.e. k ∈ N satisfies ak >

(l − ε

2

)− ε

2 = l− ε. Moreover, l+ ε2

does not satisfy this, which means that there is ε > 0 such that for ω-a.e. k ∈ N(recall that ω assumes values in 0, 1) ak 6 l+ ε

2 − ε′. Since ε′ < ε

2 must hold asotherwise l + ε

2 − ε′ < l, we conclude that l is the ultra-limit of (an)n.

The linearity of the ultra-limit operation is not hard to prove and is omitted.

For a metric space (X, d) and a point r ∈ X we say that (X, r) is a pointedmetric space. Next we want to define the ultra-limit, also called the ω-limit of asequence of pointed metric spaces (Xk, rk), relative to our fixed (in an arbitraryfashion!) non-principal ultra-filter ω.

Definition 2.2.26. Let (Xk, rk) be a sequence of pointed metric spaces. Letmoreover (xk)k∈N and (yk)k∈N be sequences of points such that xk, yk ∈ Xk foreach k ∈ N and supk∈N dXk(xk, rk) < +∞, supk∈N dXk(yk, rk) < +∞. Define thepseudo-distance

dω((xk)k, (yk)k) := limk→ω

dXk(xk, yk). (2.2.46)

44

Section 2.2

The linearity of the ω-limit of sequences of numbers proven in Lemma 2.2.25and the triangle inequality in each Xk yield the triangle inequality for the functiondω defined in (2.2.46). This pseudo-distance splits the set of all sequences (xk)k ∈Π∞k=1Xk the infinite Cartesian product of Xk’s, into equivalence classes ∼ where(xk)k ∼ (yk)k if and only if dω((xk)k, (yk)k) = 0. The ultra-limit of the sequence ofpointed metric spaces (Xk, rk) is defined to be the set of these equivalence classes,and we denote it by X ′ = limk→ω(Xk, rk). It is clear that this construction definesa metric space. A particular instance of such a construction is the ultra-limit ofa constant sequence Xk = X, rk = x ∈ X, and we call it the ultra-product of X,denoted Xω = limk→ωX. Its elements are thus equivalence classes of boundedsequences in X, and we will typically denote such elements by x, y, z, etc. For asequence (xk)k ∈ x ∈ Xω, we also write x = [(xk)] or x = limk→ω xk. It is easyto see that X ⊂ Xω by identifying each element x ∈ X with the equivalence class[(xk)k] ∈ Xω of the sequence (xk)k defined by xk = x for each k ∈ N. MoreoverX = Xω holds if and only if X is a locally compact space (see [4] and [69]).

Recall that a metric space X is said to be a length space or intrinsic space ifthe distance between any pair of points x0, x1 ∈ X equals the infimum of lengthsof all continuous curves γ[0, 1]→ X such that γ(j) = xj for j = 0, 1 (see [19] PartI Chapter 3 Section 1 for details). Clearly a geodesic space is an intrinsic space.

Theorem 2.2.27. Let ω be a non-principal ultra-filter. The ω-limit of any se-quence of metric spaces is a complete metric space. The ω-limit of a sequence ofintrinsic spaces is a geodesic space. The ω-limit of a sequence of CAT(k) spacesis a CAT(k) space.

Proof. See [19] Chapter I Lemma 5.53, Exercise 5.54 and Chapter II Corollary3.10(1), respectively.

We also need to define the ultra-extension, also called the ω-extension of aproper, l.s.c. and convex functional ϕ : X → (−∞,+∞] when X is a CAT(0)space. Observe that such an extension depends on the choice of a non-principalultra-filter, i.e. choosing a diferent one gives a functional defined on a diffetentspace. When the ultra-filter has been fixed, then we will reffer to such an extensionas to the ultra-extension of ϕ. See [4] and [69] for ultra-limits of sequences of locallyLipschitz functionals.

Definition 2.2.28. Let ω be a non-principal ultra-filter. Let X be a CAT(0)space and let ϕ : X → (−∞,+∞] be proper l.s.c. and geodesically convex. Theω-extension of ϕ is the mapping

ϕω : Xω → (−∞, ] +∞], ϕω(x) := inf limk→ω

ϕ(xk)|(xk) ∈ x (2.2.47)

By [72] Lemma 1.3 ϕ is bounded below on each ball hence ϕω(x) > −∞ foreach x ∈ Xω indeed.

Lemma 2.2.29. Let ω be a non-principal ultra-filter, let Xbe a CAT(0) space. Letmoreover ϕ : X → (−∞,+∞] be a proper, lower semi-continuous, and geodesically

45



convex functional. Then ϕω = ϕ on X and ϕω is also lower semi-continuous andgeodesically convex.

Proof. For the first claim pick x ∈ X. Then for any sequence (xk)k in the equiva-lence class of (x, x, ...) we have limk→ω d(xk, x) = 0 hence for each ε > 0, ω - a.e.k ∈ N satisfies d(xk, x) < ε. As ϕ is assumed to be l.s.c., for any δ > 0 there is ε > 0such that d(y, x) < ε implies ϕ(y) > ϕ(x) − δ. Hence for any (xk)k ∈ [(x, x, ...)]and any δ > 0 we have limk→ω ϕ(xk) > ϕ(x) − δ, and as limk→ω ϕ(x) = ϕ(x) wemust have ϕω([x, x, ...]) = ϕ(x). For the second claim let x0, x1 ∈ D(ϕω). Fixε > 0 and let (x0

k)k ∈ x0, (x1k)k ∈ x1 be such that ϕω(xj) + ε > limk→ω ϕ(xjk)

for j = 0, 1. Let moreover for k ∈ N γk : [0, 1] → X be the unique geodesicjoining x0

k and x1k. As (x0

k)k, (x1k)k are bounded sequences in X, the same holds

for (γk(t))k∈N for each t ∈ [0, 1], and we have for 0 6 s < t 6 1

dω([(γk(s))k], [(γk(t))k]) = limk→ω

d(γk(s), γk(t))

= limk→ω

(t− s)d(γk(0), γk(1)) = (t− s)dω(x0, x1)

Hence t 7→ [(γk(t))k] is a geodesic in Xω and as Xω is a CAT (0) space by Theorem2.2.27, it is the unique one connecting x0 with x1. Now we only need to observethat

ϕω([(γk(t))k]) 6 limk→ω

ϕ(γk(t)) 6(1− t) limk→ω

ϕ(x0k) + t lim

k→ωϕ(x1

k)

6(1− t)(ϕω(x0) + ε) + t(ϕω(x1) + ε)

and conclude since ε > 0 was arbitrarily chosen.

We will also use the following simple fact.

Lemma 2.2.30. Let ω be a non-principal ultra-filter. Let X be a complete metricspace, and let [(xk)] = limk→ω xk = x ∈ Xω. If for any subsequence yr = xkr of(xk)k we have limr→ω yr = x as well, then x ∈ X and xk → x in X.

Proof. The sequence (xk)k does not converge in X if and only if it is not Cauchy.Suppose that it is not Cauchy. Then for some ε > 0 for each N ∈ N we canfind k, r > N such that d(xk, xr) > ε hence we can construct two subsequences(y1r)r and (y2

r)r of (xk)k such that d(y1r , y

2r) > ε for each r ∈ N. But then

dω([(y1r)r], [(y2

r)r]) = limk→ω d(y1r , y

2r) > ε, which contradicts the assumption of

the lemma.

2.3 Construction of approximation semigroupsand some convergence theorems

The proof of the Trotter-Kato product formula on Hilbert spaces uses approxima-tion semigroups. Each of the convex functionals ϕj under consideration induces a

46

Section 2.3

contraction semigroup (Sjt )t>0, and for a discretisation parameter ρ > 0 they areused to define the operator Fρ := Skρ · · · S1

ρ . The approximation semigroups arethe semigroups generated by Fρ−I

ρ for ρ > 0. In this section, the Trotter-Kato ap-proximations are linked to the approximation semigroups, and the approximationsemigroups are linked to the semigroup (St)t>0 generated by the sum functionalϕ1 + · · ·+ϕk. The final result of this section is the convergence of the Trotter-Katoapproximations, provided the resolvents of the approximation semigroups convergeto the resolvents of (St)t>0.

In this section we will first recall the strategy of proving the product formula(2.2.45) on a Hilbert space. Then we will point out why it is reasonable to expectthat it will work in a CAT(0) space as well. After that we will prepare the toolsthat we need in the next section. As we already mentioned in the introduction, theresults we obtain in this section are of independent interest. Although our analysisroughly follows the program of the proof in the Hilbert spaces setting, there aremany aspects which require new ideas and techniques. First, the definition ofresolvents associated to approximation semigroups uses linear convexity which wereplace by geodesic convexity. Secondly, the Hilbert space proof uses integralsof Hilbert space valued curves, where we have to use a suitable discretizationinstead. Thirdly, our construction of the approximation semigroups is carried outby mimicking the Crandall-Liggett approach of doubling the variables.

Let us start our analysis by recalling some details of the classical proof. To fixideas, consider two convex, proper, l.s.c functionals ϕ1, ϕ2 defined on a Hilbertspace H, and let us discuss the version of the product formula where both stepsare taken by the resolvents, i.e. in Theorem 2.2.22 we take Ft := J 2

t J 1t . Note the

following facts. First of all for any contraction F on a closed convex subset C ofH and for any ρ > 0 we can construct the semigroup

(Sρ,Ft

)t>0

of contractions

on C whose paths are the solutions of

Sρ,Ft x = x−∫ t

0

(I − F )ρ

Sρ,Fs x ds for x ∈ C. (2.3.1)

Such semigroups can be constructed by a fixed-point argument in a suitablespace of curves. Alternatively one can observe that I−F

ρ is a maximal monotoneoperator and use the results of [17]. Next, since our nonlinear generator F−I

ρ isLipschitz and we are in a Hilbert space these solutions can also be obtained bythe Euler forward procedure. That is, we can solve the discrete version of ourequation (2.3.1) with a uniform time step size τ as follows: for an initial conditionx0 find xτ such that

xττ − x0

τ=F − Iρ

x0,

which is

xττ = x0 +τ

ρ(F − I)x0, (2.3.2)

47



and define further by induction xτkτ := xτ(k−1)τ + τρ (F − I)xτ(k−1)τ . Then it is not

hard to show that these discrete solutions converge to the solution of (2.3.1). Wecan do this for any contraction F , in particular for F := Fρ for any ρ ∈ (0,+∞),and let us write Sρt := S

ρ,Fρt . Then for m ∈ N and t ∈ (0,+∞) and the choice

ρ = τ := t/m the above described Euler forward approximation of (2.3.1) readsxτkτ = (Ft/m)k and in particular xt/mt = (Ft/m)mx0, i.e. the Trotter approximationof Stx0. This fact clarifies why we are able to find a good estimate betweensolutions of (2.3.1) and the Trotter approximations (see Lemma 2.3.11 below forthe precise statement and the proof in our setting).

The classical proof argues further in several steps. First, one shows that in thissituation pointwise convergence of the resolvents

Jλ,ρ :=(I +

λ

ρ(I − Fρ)

)−1

ρ > 0 (2.3.3)

of the approximation semigroups (Sρt )t (recall (2.2.43)), to the resolvent Jλ of(St)t>0 defined in (2.2.26) as ρ → 0 for each λ > 0 implies convergence of thesemigroups. Second, one deduces an estimate of the distance between the semi-groups (Sρt )t>0 and the Trotter iterations in (2.2.45)—this estimate seems naturalin light of our informal analysis (we have proved the same type of estimate in theCAT(0) setting in (2.3.11)). Third, one proves that convergence of the resolventsindeed holds (which is done in [58]). Now a careful reader of the classical proofsmight notice that this part of the argument essentially works due to convexityof the Hilbert norm and the Lipschitz property of Ut. Thus since we have theseproperties in the CAT(0) setting too, we may hope to be able to use a similarstrategy.6

Now that we know what to do, let us construct the approximation semigroupsas above but in the CAT(0) setting. We will do so by constructing the resolvents(2.3.3) first, then the we obtain similar estimates as in the work of Crandall-Liggett[30], and finally mimic their proof of convergence. Observe that by constructionthese approximation semigroups indeed coincide with the classical ones if X is aclosed convex subset of a Hilbert space.

In the Hilbert space case the resolvents (I + λρ (I − Fρ))−1 play an important

role. In order to define such an operator in our case of CAT(0) spaces, we consideran equivalent formulation which uses linear convex combinations and replace thatby geodesic convex combinations.

If X is a subset of a linear space, then the identity (I + λρ (I − F ))−1x = y is

equivalent to y = 11+λ

ρ

x +λρ

1+λρ

Fy, that is y is the unique point in X having the

property to be the point γ( λ/ρ1+λ/ρ ) on the geodesic γ connecting x and Fy. Hence

6We may however not hope to have a theorem of such generality for gradient flows on spaceswith convex metric as even on Banach spaces, already for the existence of the flow one needs acompactness assumption on the lower level sets.

48

Section 2.3

we are led to define the mapping

Gλ,ρ,x : X → X, Gλ,ρ,x(y) :=1

1 + λρ

x⊕λρ

1 + λρ

Fy (2.3.4)

for a contraction F : X → X (but we omit F in the notation for simplicity).

Lemma 2.3.1. Let F : X → X be a contraction. Then for any λ, ρ > 0 andx ∈ X the mapping Gλ,ρ,x is a strict contraction with the Lipschitz constant l 6λ/ρ

1+λ/ρ < 1. Consequently for each λ, ρ > 0 and x ∈ X the mapping Gλ,ρ,x has aunique fixed point denoted Jλ,ρx which moreover satisfies

Jλ,ρx =1

1 + λ/ρx⊕ λ/ρ

1 + λ/ρFJλ,ρx. (2.3.5)

Proof. Fix λ, ρ > 0 and x ∈ X. Proposition 2.2.8 and the contraction property ofF yield that for y1, y2 ∈ X

d(Gλ,ρ,x(y1), Gλ,ρ,x(y2)) 6λ/ρ

1 + λ/ρd(Fy1, Fy2) 6

λ/ρ

1 + λ/ρd(y1, y2)

which is the first assertion. Now by the Banach fixed point theorem Gλ,ρ,x indeedhas a unique fixed point, and (2.3.5) holds due to (2.3.4).

Definition 2.3.2. Let (X, d) be a CAT(0) space and let F : X → X be a contrac-tion. Then for each λ, ρ > 0 the mapping Jλ,ρ : X 3 x 7→ Jλ,ρx defined by Lemma2.3.1 is called the (λ, ρ) resolvent associated to F . If the parameters λ and ρ arenot specified then we will refer to these mappings as the resolvents associated toF . For λ = 0 and ρ > 0 we define Jλ,ρ := I, the identity map X → X.

Notice that this definition coincides with(I + λ

ρ (I − F ))−1

whenX is a convexsubset of a Hilbert space.

Lemma 2.3.3. Let F : X → X be a contraction. Then for each λ, ρ > 0 the mapJλ,ρ := J Fλ,ρ : X → X is a contraction.

Proof. Let t := λ/ρ1+λ/ρ ∈ (0, 1) and x1, x2 ∈ X. Proposition 2.2.8 gives

d(Jλ,ρx1,Jλ,ρx2) 6 (1− t)d(x1, x2) + td(FJλ,ρx1, FJλ,ρx2)6 (1− t)d(x1, x2) + td(Jλ,ρx1,Jλ,ρx2)

hence d(Jλ,ρx1,Jλ,ρx2) 6 d(x1, x2).

The following lemma requires a different proof than in the linear case, whichuses monotonicity properties of I − F .

49



Lemma 2.3.4. Let F : X → X be a contraction and let Jλ,ρ := J Fλ,ρ. Then forλ, ρ > 0, x ∈ X we have

d(x,Jλ,ρx)λ

6d(x, Fx)

ρ. (2.3.6)

Proof. Since Jλ,ρx is the unique fixed point of the mapping Gλ,ρ,x, we have thatlimn→+∞(Gλ,ρ,x)n(x) = Jλ,ρx. By Lemma 2.3.1 we estimate

d(x,Jλ,ρx) 6∞∑n=1

d((Gλ,ρ,x)n−1(x), (Gλ,ρ,x)n(x))

6∞∑n=1

(λ/ρ

1 + λ/ρ

)n−1

d(x,Gλ,ρ,x(x))

61

1− λ/ρ1+λ/ρ

d(x,Gλ,ρ,x(x)) =λ

ρd(x, Fx)

since by definition d(x,Gλ,ρ,x(x)) = λ/ρ1+λ/ρd(x, Fx).

Lemma 2.3.5. (The Resolvent Identity) Let F : X → X be a contraction, andlet Jλ,ρ := J Fλ,ρ. Then for each x ∈ X, 0 < µ < λ, ρ > 0 the Resolvent Identityholds:

Jλ,ρx = Jµ,ρ(µ

λx⊕ λ− µ

λJλ,ρx

)(2.3.7)

Proof. For any y ∈ X we have by definition that Jµ,ρy is the unique point z ∈ Xthat has the property that

d(y, z) =µ/ρ

1 + µ/ρd(y, Fz). (2.3.8)

Hence it is enough to show that for y := µλx⊕

λ−µλ Jλ,ρx equality (2.3.8) holds for

z := Jλ,ρx. To this aim set a := d(x, FJλ,ρx), b := d(Jλ,ρx, FJλ,ρx) = 11+λ/ρa =

ρρ+λa, c := d(y,Jλ,ρx) = (a− b)µλ = λ

ρ+λ ·µλa = µ

ρ+λa. If d(y, FJλ,ρx) = b+ c = 0holds then since λ, ρ > 0 we must have that a = 0 too and x = Jλ,ρx = FJλ,ρx =Fx and this implies that for each µ > 0 we also must have x = J µ, ρx so that(2.3.7) follows trivially. Thus we may assume that d(y, FJλ,ρx) = b + c 6= 0. Aneasy computation gives

d(y,Jλ,ρx)d(y, FJλ,ρx)

=c

b+ c=

µρ+λaρ+µρ+λa

=µ

ρ+ µ,

which is (2.3.8), and we have completed the proof.

Lemma 2.3.6. Let F : X → X be a contraction, and let Jλ,ρ := J Fλ,ρ. Then forλ, ρ > 0, x ∈ X and n ∈ N

d(J nλ,ρx, x) 6 nλ

ρd(x, Fx). (2.3.9)

50

Section 2.3

Proof. By the triangle inequality, Lemma 2.3.3 and Lemma 2.3.4 we estimate

d(Jλ,ρx, x) 6n∑j=1

d(J jλ,ρx,Jj−1λ,ρ x) 6 nd(Jλ,ρx, x) 6 n

λ

ρd(x, Fx). (2.3.10)

The proof of the following lemma relies on the convexity of CAT(0) distances,i.e. (2.2.16).

Lemma 2.3.7. Let F : X → X be a contraction, and let Jλ,ρ := J Fλ,ρ. Then forρ > 0, 0 < µ 6 λ and n,m ∈ N, n > m we have

d(J nµ,ρx,Jmλ,ρx) 6m−1∑j=1

αjβn−1B(n, j)d(Jm−jλ,ρ x, x)+

+n∑

j=m

αmβj−mB(j − 1,m− 1)d(Jm−jµ,ρ x, x)

(2.3.11)

where α = µλ , β = λ−µ

λ and B(·, ·) are the binomial coefficients.

Proof. Fix ρ, µ, λ, x as above and let n > m. For integers j and k such that0 6 j 6 n, 0 6 k 6 m we define ak,j := d

(J jµ,ρx,J kλ,ρx

). With the aid of Lemma

2.3.4, Lemma 2.3.5 and (2.2.16) we estimate

ak,j = d

(J jµ,ρx,Jµ,ρ

(µ

λJ k−1λ,ρ x⊕ λ− µ

λJ kλ,ρx

))6d

(J j−1µ,ρ x,

µ

λJ k−1λ,ρ x⊕ λ− µ

µJ kλ,ρx

)6µ

λd(J j−1µ,ρ x,J k−1

λ,ρ x)

+λ− µµ

d(J j−1µ,ρ x,J kλ,ρx

)= αak−1,j−1 + βak,j−1.

This is precisely the same estimate as in [30] Lemma 1.3 and we conclude the theclaim.

Lemma 2.3.8. Let F : X → X be a contraction, and let Jλ,ρ := J Fλ,ρ. Then forρ > 0, < µ 6 λ, n,m ∈ N, n < m we have

d(J nµ,ρx,Jmλ,ρx

)6

n∑j=0

αjβn−jB(n, j)d(Jm−jλ x, x

). (2.3.12)

Proof. As in previous lemma we obtain precisely the same estimates as in [30]hence we may conclude our claim.

The following Lemma is from [30] and we state it for the completeness.

51



Lemma 2.3.9. Let n > m > 0 be integers, and α and β positive numbers addingto 1. Then

1.∑nj=0 B(n, j)αjβn−j(m− j) 6

√(nα−m)2 + nαβ

2.∑nj=m B(j − 1,m− 1)αmβj−m(n− j) 6

√mβα2 +

(mβα +m− n

)2

Proof. See [30] Lemma 1.4.

We are now able to construct the approximation semigroup. The approach byMiadera-Oharu [73] (also to be found in Brezis [17]) uses Bochner integration andtherefore seems inappropriate to generalize to CAT(0) spaces. Instead, we willmimic the Crandall-Liggett approach.

Theorem 2.3.10. Let X be a complete CAT(0) space and let F : X → X be acontraction. Then denoting Jλ,ρ := J Fλ,ρ and (Sρt )t>0 :=

(Sρ,Ft

)t>0

we have that

for each ρ > 0 the exponential formula holds:

Sρt x := limn

(J tn ,ρ

)nx (2.3.13)

exists for all t > 0 and all x ∈ X. Moreover (Sρt )t>0 is a contraction semigroupon X with globally Lipschitz paths, and we have the following two estimates

d (Sρt x, Sρsx) 6 2

d(x, Fx)ρ

|t− s|, ∀x ∈ X, ∀ t, s > 0, (2.3.14)

and

d(Sρsx,J nt/n,ρx

)6 2t

1√n

d(x, Fx)ρ

. (2.3.15)

Proof. Let x ∈ X, ρ > 0. We abbreviate Jλ := Jλ,ρ throughout this proof. Letλ > µ > 0, and let n,m ∈ N with n > m. By Lemma 2.3.7, Lemma 2.3.6 and

52

Section 2.3

Lemma 2.3.9 we estimate

d(J nµ x,Jmλ x

)6m−1∑j=0

αjβn−jB(n, j)d(Jm−jλ x, x

)+

n∑j=m

αmβj−mB(j − 1,m− 1)d(Jm−jµ x, x

)6λ

m−1∑j=0

αjβn−jB(n, j)(m− j)d(x, Fx)ρ

+

+ µ

n∑j=m

αmβj−mB(j − 1,m− 1)(m− j)d(x, Fx)ρ

6[λ

√(nµ

λ−m

)2

+ nµ

λ

λ− µλ

+ µ

√λ2

µ2

λ− µλ

m+(λ

µ

λ− µλ

m+m− n)2

]d(x, Fx)

ρ

= [√

(nµ− λm)2 + nµ(λ− µ)

+√mλ(λ− µ) + (mλ− nµ)2]

d(x, Fx)ρ

.

(2.3.16)

For µ := tn , λ := t

m the inequality above reads

d(Jmt/mx,J

nt/nx

)6 2t

√∣∣∣∣ 1m− 1n

∣∣∣∣d(x, Fx)ρ

. (2.3.17)

Hence by completeness of X the limit

Sρt x := limn

(Jt/n

)nx (2.3.18)

exists for t > 0 and x ∈ X. Moreover taking the limit m → +∞ in (2.3.17) weobtain the error estimate (2.3.15). We define Sρ0 to be the identity map on X.As Jt/n = Jt/n,ρ is a contraction for all t > 0 and n ∈ N, (Sρt )t>0 is a familyof contractions. Moreover choosing n = m, µ := t/n and λ := s/n in (2.3.16)and passing to the limit gives the estimate (2.3.14) of the theorem, and the globalLipschitz property of the paths.

Next we show the semigroup property. To this aim let m ∈ N, t > 0 and x ∈ X.Since J nt/n = J nt/n,ρ is a contraction and (2.3.16) holds,

d(

(Sρt )m x,(J nt/n

)mx)

6d(Sρt (Sρt )m−1

x,J nt/n (Sρt )m−1x)

+

+ d(J nt/n (Sρt )m−1

x,J nt/n(J n(m−1)t/n

)x)→ 0

53



as n→∞. Hence for ρ > 0, t 6 0 and m ∈ N we have that

(Sρt )m x := limn

(J nt/n

)mx = lim

nJ nmmtnm

x = Sρmtx. (2.3.19)

So for l, k, r,m ∈ N (2.3.19) gives

Sρlk+ n

m

= Sρlm+nkkm

=(Sρ1km

)lm+nk

=(Sρ1km

)lm (Sρ1km

)nk= Sρlm

km

Sρnkkm

= Sρl/kSρr/m,

i.e. Sρt+s = Sρt Sρs for s, t ∈ Q+. Finally for arbitrary s, t > 0 take two sequences

of positive rational numbers sn → s, tn → t. We just showed that StnSsnx =Stn+snx. Observe moreover that (2.3.14) implies Sρtn+sn → Sρt+s for x ∈ X. Wemay conclude the that the semigroup property holds by the following estimate:

d(SρtnSρsnx, S

ρt S

ρsx) 6d(SρtnS

ρsnx, S

ρtnS

ρsx) + d(SρtnS

ρsx, S

ρt S

ρsx)

6d(Sρsnx, Sρsx) + d(SρtnS

ρsx, S

ρt S

ρsx)→ 0 as n→ +∞.

In the Hilbert space case the non-linear generator of the equation (2.3.1) is I−Fρ

a maximal monotone operator, and for each x in the domain of the semigroup theLipschitz constant of the corresponding path is just |x−Fx|ρ . We however appliedthe resolvent identity to construct these semigroups in our CAT(0) space case,which means that we used the convexity of the metric (2.2.16) in our estimates,instead of using the (−1) convexity of its square. This results in the constant2 at the right side of (2.3.14) and also in the estimates in the following lemma.However, this is immaterial in our considerations.

We are going to need the following estimate (see [17] Theoreme 1.7 and [73]for the linear version of this estimate). The proof for the linear case uses Bochnerintegration and a Gronwall type argument. By means of suitable characterizationswe are able to give a proof in our setting.

Lemma 2.3.11. Let F : X → X be a contraction and let (Sρt )t>0 be the semigroupconstructed in Theorem 2.3.10 for ρ > 0. Then for t > 0, x ∈ X and m ∈ N wehave the following:

1. Sρt x = S1t/ρx

2. d(S1t x, F

mx) 6 2√

(m− t)2 + td(x, Fx)

3. d(Sρt x, Fmx) 6 2

√(m− t

ρ

)2

+ tρd(x, Fx).

54

Section 2.3

Proof. The last claim is a direct consequence of the first two. The first claimfollows by the simple observation that for all ρ, λ > 0, Jλ,ρ = Jλ

ρ ,1, hence for

x ∈ X

Sρt xn→∞←− J nt

n ,ρx = J nt/ρ

n ,1xn→∞−→ S1

(t

ρ

)x.

The second claim requires more effort. Recall that for each y ∈ X Jλx := Jλ,1x =1

1+λy ⊕λ

1+λFJλ,1y. Hence by (2.2.2)

d2(Fmx,J ntnx) 6

11 + t

n

d2(Fmx,J n−1tn

x) +tn

1 + tn

d2(Fmx, FJ n−1tn

x).

Repeating the argument n times we obtain

d2(Fmx,J ntnx) 6

(1 +

t

n

)−nd2(Fmx, x)

+t

n

n∑k=1

(1 +

t

n

)−kd2(Fmx, FJ n−kt

n

x)

6

(1 +

t

n

)−nm2d2(Fx, x)

+t

n

n∑k=1

(1 +

t

n

)−kd2(Fm−1x,J n−kt

n

x)

=(

1 +t

n

)−nm2d2(x, Fx)

+t

n

n∑k=1

(1 +

t

n

)−(n−k)

d2(Fm−1x,J ktnx)

For n ∈ N define functions

fn(s) :=n∑k=1

(1 +

t

n

)−(n−k)

1( (k−1)tn , ktn ](s),

gn(s) :=n∑k=1

d2(Fm−1x,J ktnx)1( (k−1)t

n , ktn ](s).

With this notation the above inequality becomes

d2(Fmx,J ntnx) 6

(1 +

t

n

)−nm2d2(x, Fx) +

∫ t

0

fn(s)gn(s)ds.

55



We are going to show that fn(s) → e−(t−s) and gn(s) → d2(Fm−1x, Ssx) fors ∈ (0, t] and that supn∈N,s∈[0,t] |gn(s)||fn(s)| < ∞. Then by the dominatedconvergence theorem we will have

d2(Fmx, Stx) 6 e−tm2d2(x, Fx) +∫ t

0

e−(t−s)d2(Fm−1x, Ssx)ds (2.3.20)

(we abbreviate Ss := S1s ). Let us establish these facts.

Pick n ∈ N and s ∈ (0, t]. There is a unique 0 < ks,n 6 n such that (ks,n−1)tn <

s 6 ks,ntn .

Filling in ks,n, tn ,m, sm for n,µ,m,λ, respectively, in (2.3.16) gives

d(J ks,ntn

x,Jmsmx)

6 (

√(ks,nt

n− s)2

+ks,nt

n

(s

m− t

n

)+

+

√s

(s

m− t

n

)+(ks,nt

n− s)2

)2d (x, Fx)

ρ(2.3.21)

and taking limit for m→∞ gives

d(J ks,ntn

x, Ssx)

6 (

√(ks,nt

n− s)2 +

ks,nt

n

t

n+

√st

n+ (

ks,nt

n− s)2)

2d(x, Fx)ρ

(2.3.22)In light of (2.3.16) it is not hard to see that

J ktnx|n ∈ N, 0 6 k 6 n

is a

bounded subset of X, hence the identity∣∣gn(s)− d2(Fm−1x, Ssx)∣∣ =

∣∣∣(d(Fm−1x,J ks,ntn

x)

+ d(Fm−1x, Ssx

))·(d(Fm−1x,J ks,nt

n

x)− d

(Fm−1x, Ssx

))∣∣∣ ,the triangle inequality and the estimate (2.3.22) now easily yield that

gn(s)→ d2(Fm−1x, Ssx)

for s ∈ (0, t] as well as that gn is a uniformly bounded sequence of functions.The claims concerning fn are easy to handle, and we omit the details. Hence weestablished (2.3.20). Now if d(x, Fx) = 0, then clearly Jλ,ρ = x for all λ, ρ > 0thus also Sρsx = x for all s > 0 and ρ > 0 and the estimate in item 2 followstrivially. Assume that d(x, Fx) > 0. To complete the proof set

ϕm(s) :=d(Fmx, Ssx)

2d(x, Fx)for m > 0 and s > 0 (2.3.23)

so that (2.3.20) reads

ϕ2m(t) 6

12e−tm2 +

∫ t

0

e−(t−s)ϕ2m−1(s)ds (2.3.24)

56

Section 2.3

for m > 1 (F 0 being the identity map by definition). We have to show that

ϕ2m(t) 6 ((m− t)2 + t), ∀ t > 0 (2.3.25)

and we will do that by induction.By (2.3.14) we have that ϕ0(t) 6 t, thus (2.3.25) holds for m = 0. Suppose the

claim is true for m− 1. Then by (2.3.24)

ϕ2m(t) 6

12e−tm2 +

∫ t

0

e−(t−s)((m− 1− s)2 + s)ds

hence showing

m2 +∫ t

0

es((m− 1− s)2 + s)ds = et((m− t)2 + t)

suffices. Well, at t = 0 both sides coincide and the derivatives of the two functionscoincide as well:

et((m− 1− t)2 + t) = et((m− t)2 + t)− 2(m− t)et + et =d

dtet((m− t)2 + t)

Now the proof is complete.

Next we consider a finite sequence ϕ1, ..., ϕk of geodesically convex l.s.c func-tionals X → (−∞,+∞] such that ϕ :=

∑kj=1 ϕj 6≡ +∞. Then Assumption 2.2.13

holds for each of these functionals, i.e. each ϕj generates a contraction semigroup(Sjt )t>0 on D(ϕj). Moreover Xj = D(ϕj) is a closed geodesically convex subset ofX and we denote Pj : X → Xj to be the nearest point projection onto Xj as inProposition 2.2.10. Then for t > 0, j = 1, 2, ..., k and x, y ∈ X

d(SjtPjx, S

jtPjy

)6 d(Pjx, Pjy) 6 d(x, y). (2.3.26)

Define mappings

F 1t : X → X, F 1

t := Skt Pk · · · S1t P1, t > 0. (2.3.27)

By (2.3.26) F 1t is a contraction for each t > 0. Recalling (2.2.26) we further define

mappingsF 2t : X → X, F 2

t := J kt · · · J 1t . (2.3.28)

Due to Proposition 2.2.15, F 2t is also a contraction for each t > 0.

Theorem 2.3.12. Let either Fρ = F 1ρ for each ρ > 0 or Fρ = F 2

ρ for each ρ > 0.Let moreover for ρ > 0 (Sρt )t>0 be the semigroup constructed in Theorem 2.3.10for the choice F = Fρ, and let Jλ,ρ be its resolvents. Denote Jλ to be the resolventsof the sum functional ϕ :=

∑kj=1 ϕj, and (st)t>0 the semigroup associated to ϕ.

Then convergence of the resolvents

Jλ,ρx→ Jλx as ρ ↓ 0, ∀x ∈ D(ϕ), ∀λ > 0, (2.3.29)

57



implies thatSρt x→ Stx for x ∈ D(ϕ) (2.3.30)

and for each x ∈ D(ϕ), the limit is uniform on compact time intervals.

Proof. Fix T > 0, x ∈ D(|∂ϕ|) and 0 < t 6 T . For each λ > 0 our assumptionimplies

d(x,Jλ,ρ)λ

−→ d(x,Jλx)λ

6 |∂ϕ|(x) (2.3.31)

as ρ ↓ 0 where the inequality holds by (2.2.32). Next we have the followingestimates:

d(Sρt x, Stx) 6d(Sρt x, Sρt Jλ,ρx) + d(Sρt Jλ,ρx, Stx)

6d(x,Jλ,ρx) + d(Sρt Jλ,ρx, Stx)(2.3.32)

and

d(Sρt Jλ,ρx, Stx) 6d(Sρt Jλ,ρx, (Jt/n,ρ)nJλ,ρx)+ d((Jt/n,ρ)nJλ,ρx, (Jt/n,ρ)nx)+ d((Jt/n,ρ)nx, (Jt/n)nx) + d((Jt/n)nx, Stx).

(2.3.33)

We need to find upper bounds for the four expressions appearing in (2.3.33).Firstly by (2.3.15) we have

d(Sρt Jλ,ρx, (Jt/n,ρ)nJλ,ρx) 62t√nd(Jλ,ρx, FρJλ,ρx) =

2t√n

ρ

λd(x,Jλ,ρx) (2.3.34)

where the equality follows by (2.3.5). As Jλ,ρ is a contraction, we have that foreach n ∈ N,

d((J tn ,ρ

)nJλ,ρx, (J tn ,ρ

)nx) 6 d(Jλ,ρx, x) (2.3.35)

and by the estimate (2.2.37), we have

d((J tn

)nx, Stx

)6

t√2n|∂ϕ|(x). (2.3.36)

Well now for this arbitrary x ∈ D(|∂ϕ|), n ∈ N, t 6 T , and λ > 0 we haveobtained the following estimate

d (Sρt x, Stx) 62d(x,Jλ,ρx) +2t√n

ρ

λd(x,Jλ,ρx)

+d(

(J tn ,ρ

)nx, (J tn

)nx)

+t√2n|∂ϕ|(x).

(2.3.37)

To show convergence for x ∈ D(|∂ϕ|) let ε > 0, and choose λ0 > 0 so thatλ0|∂ϕ|(x) < ε.

By (2.3.31) there is a δ > 0 such that for ρ < δ

d (x,Jλ0,ρx)λ0

6d (x,Jλ0x)

λ0+ ε

58

Section 2.3

henced (x,Jλ0,ρx) 6 λ0|∂ϕ|(x) + ελ0 < 2ε for ρ < δ. (2.3.38)

Next fix n0 ∈ N such that

t√2n0

|∂ϕ|(x) 6T√2n0

|∂ϕ|(x) < ε. (2.3.39)

Let ρ1 := min√n0ελ0

2T (λ0|∂ϕ|(x)+ελ0) , δ. Then for ρ < ρ1 by (2.3.38) we estimate

2t√n0

ρ

λ0d (x,Jλ0,ρx) 6

2T√n0

ρ0

λ0(λ0|∂ϕ|(x) + ελ0) 6 ε. (2.3.40)

Finally for t 6 T we estimate

d(

(J tn0,ρ)

n0x, (J tn0

)n0x)

6 d(

(J tn0,ρ)

n0x, (J tn0,ρ)

n0−1J tn0x)

(2.3.41)

+ d(

(J tn0,ρ)

n0−1J tn0x, (J t

n0)n0−1J t

n0x),

hence the assumption (2.3.29) and induction give existence of ρ0 6 ρ1 such thatfor ρ < δ0

d((Jt/n0,ρ

)n0x,(Jt/n0

)n0x)< ε (2.3.42)

holds. Now (2.3.37), (2.3.39), (2.3.40), (2.3.41) and (2.3.42) give

limρ→0

Sρt x = Stx for t > 0 and x ∈ D(|∂ϕ|). (2.3.43)

To show that for arbitrary but fixed x ∈ D(|∂ϕ|) and T > 0 the convergencein (2.3.43) is uniform for t 6 T , pick τ ∈ (0, T ). Applying the triangle inequality,contraction property of Sρt and (2.3.14) we obtain

d(Stx, Sρt x) 6d (Stx, S

ρt Jλ,ρx) + d (Sρt Jλ,ρx, S

ρt x)

6d (Stx, Sτx) + d (Sτx, SρτJλ,ρx)+d (SρτJλ,ρx, S

ρt Jλ,ρx) + d (Jλ,ρx, x)

6|t− τ ||∂ϕ|(x) + d (Sτx, SρτJλ,ρx)

+|t− τ |d(Jλ,ρx, FρJλ,ρx)ρ

+ d(Jλ,ρx, x).

(2.3.44)

Now observe the following facts. We can fix a small λ > 0 such that for suffi-ciently small ρ > 0 the quantity d(Jλ,ρx, x) becomes arbitrarily small, as we didin (2.3.38). Furthermore, we have that d(Jλ,ρx,FρJλ,ρx)

ρ = d(x,Jλ,ρx)λ and for any

fixed λ this term is bounded as ρ ↓ 0. We already argued in (2.3.33) that for eachτ 6 T and λ > 0, the quantity d(Sτx, SρτJλ,ρx) is small for small ρ. Well nowwe can finish the argument by the compactness of [0, T ] and conclude that forx ∈ D(|∂ϕ|) the convergence in (2.3.43) is uniform on [0, T ].

59



At last for x ∈ D(ϕ) = D(|∂ϕ|) pick ε > 0, and x ∈ D(|∂ϕ|) such thatd(x, y) < ε and estimate

d (Sρt x, Stx) 6d (Sρt x, Sρt y) + d (Sρt y, Sty) + d (Sty, Stx)

62ε+ d (Sρt y, Sty) .(2.3.45)

The proof is now complete.

We conclude this section with the following theorem.

Theorem 2.3.13. Let either Fρ = F 1ρ for each ρ > 0 or Fρ = F 2

ρ for each ρ > 0.Let moreover for ρ > 0 (Sρt )t>0 be the semigroup constructed in Theorem 2.3.10 forthe choice F = Fρ, and let Jλ,ρ be its resolvents. Denote Jλ to be the resolventsof the sum functional ϕ :=

∑kj=1 ϕj, and (St)t>0 the semigroup associated to

ϕ. Then the convergence of the resolvents in (2.3.29) implies that the followingextension of the Trotter-Kato product formula holds:(

F tn

)nx −→ Stx, ∀x ∈ X. (2.3.46)

Moreover, for each x ∈ D(ϕ) the limit in (2.3.17) is uniform on compact timeintervals.

Proof. Assume first that x ∈ D(|∂ϕ|). Fix T > 0 and let t 6 T . We estimate

d(Stx,

(F tn

)nx)

6 d(Stx, S

tnt x)

+ d(Stnt x,

(F tn

)nx)

(2.3.47)

and handle the two terms separately. For ρ > 0, n ∈ N and λ > 0 we have

d(Sρnρx, (Fρ)

nx)

6d(Sρnρx, S

ρnρJλ,ρx

)+

+ d(SρnρJλ,ρx, (Fρ)nJλ,ρx

)+ d ((Fρ)nJλ,ρx, (Fρ)nx)

62d(x,Jλ,ρx) + d(SρnρJλ,ρx, (Fρ)nJλ,ρx

),

where the equality above follows directly by (2.3.5). Next for ρ = tn and λ > 0 we

have by Lemma 2.3.11 that

d(SρnρJλ,ρx, (Fρ)nJλ,ρx

)62

√(n− nρ

ρ

)2

+nρ

ρd (Jλ,ρx, FρJλ,ρx)

= 2√nρd(x,Jλ,ρx)

λ

62T√n

d(x,Jλ,ρx)λ

.

Choose ε > 0 and pick 1 > λ0 > 0 such that d(x,Jλ0x) 6 λ0|∂ϕ|(x) < ε (the firstinequality holds by (2.2.32)). Observe that for ρ > 0 small enough 2d(x,Jλ,ρx) 62d(x,Jλ0x)+ε < 3ε holds due to our assumption, and then for n large enough the

60

Section 2.4

same holds for ρ = tn for any t ∈ [0, T ]. Since λ0 is fixed, we have the convergence

d(x,Jλ,t/nx)

λ0→ d(x,Jλ0x)

λ0< +∞ as ρ ↓ 0, hence for any t ∈ [0, T ] one can find n0

so that for n > n0, T√n

d(x,Jλ0,t/nx)

λ0< ε holds. Now (2.3.47) and Theorem 2.3.12

yield (2.3.46) uniformly on compact time intervals for x ∈ D(|∂ϕ|). For arbitraryx ∈ D(ϕ) let y ∈ D(|∂ϕ|) be such that d(x, y) < ε to estimate

d(Stx, Fnt/nx) 6 2d(x, y) + d(Sty, Fnt/ny)

and conclude by the previous.

2.4 The Trotter product formula

In this section we prove the Trotter-Kato product formula in CAT(0) spaces (The-orem 2.4.4) and several related versions. Due to Theorem 2.3.13 it suffices toestablish the convergence of the approximate resolvents in (2.3.29).

The classical proof by Kato-Masuda of the convergence of approximate resol-vents uses the notion of weak convergence in Hilbert spaces in an essential way, inparticular weak sequential compactness of bounded sets. In CAT(0) spaces thereappears to be no suitable topology that replaces the role of weak convergence anda new idea is needed. It turns out that the ultra-limits of bounded sequences canbe used to complete the proof in the case of CAT(0) spaces.

The following sequence of lemmas establishes required estimates. These lem-mas are inspired by the lemmas in [58] that establish similar estimates—argumentsrelying on the inner product structure are replaced by suitable arguments that canbe given with the aid of the CAT(0) inequality (2.2.2).

We will use the same notation as in the previous section, in particular (2.3.27)and (2.3.28). Furthermore we denote

U jt := Sjt Pj , t > 0, j = 1, ..., k , (2.4.1)

orU jt := J jt , t > 0, j = 1, ..., k , (2.4.2)

unless we specify.

Lemma 2.4.1. Let z ∈ D(ϕ). Set z0(t) = z, and zj(t) := (U jt · · · U1t )z for

j = 1, 2, ..., k and for t > 0. Then there is a constant c∗ > 0 such that

d(zj(t), z) 6 c∗√t , t ∈ (0, 1), j = 1, 2, ..., k (2.4.3)

Proof. For j = 1, 2, ..., k we have

d(zj(t), z) 6 d(zj(t), Ujt z) + d(U jt z, z) 6 d(zj−1(t), z) + d(U jt z, z)

hence by induction we only need to estimate d(U jt z, z). Since by definition D(ϕ) ⊂D(ϕj) for each j (2.2.39) gives

d2(U jt z, z) + 2tϕj(Ujt z) 6 2tϕj(z)

61



so by (2.2.24)

d2(U jt z, z

)+ 2tbd

(U jt z, z

)+ 2t(c− ϕj(z)) 6 0 (2.4.4)

holds, where b, c ∈ R depend only on z. Hence we must have that d(U jt z, z

)is

bounded by the positive root of the quadratic equation in (2.4.4), i.e.

d(U jt z, z

)6−2tb+

√4t2b2 − 8t(c− ϕj(z))

2(2.4.5)

and since there are finitely many j’s to be considered while t <√t holds if t ∈ (0, 1),

we obtain existence of c∗ such that (2.4.3) holds.

Let us fix x ∈ X and λ > 0 in what follows. Define for t > 0

x0(t) := Jλ,tx (2.4.6)

xj(t) := U jt xj−1(t) for j = 1, 2, ..., k. (2.4.7)

Observe that xk(t) = Ftx0(t) hence by construction (recall (2.3.4) and Lemma2.3.1) x0(t) = 1

1+λ/tx⊕λ/t

1+λ/txk(t) thus also

d(x, xk(t)) =1 + λ/t

λ/td(x, x0(t)) =

t+ λ

λd(x, x0(t)) (2.4.8)

for each t > 0. Let us also rewrite (2.2.2) for arbitrary base point v ∈ X as

d2(v, x0(t))− 11 + λ/t

d2(v, x) +λ/t

1 + λ/t

11 + λ/t

d2(x, xk(t)) 6λ/t

1 + λ/td2(v, xk(t)).

(2.4.9)If Ft is given by (2.3.27) and U jt by (2.4.1), then by (2.2.40) we have

12td2(xj(t), v)− 1

2td2(xj−1(t), v) + ϕj(xj(t)) 6 ϕj(v) ∀v ∈ D(ϕj) (2.4.10)

for t > 0 and j = 1, 2, ..., k. By adding these variational inequalities we obtain

12td2(xk(t), v)− 1

2td2(x0(t), v) +

k∑j=1

ϕj(xj(t)) 6 ϕ(v) (2.4.11)

for each for each t > 0 and v ∈ D(ϕ). On the other hand if Ft is given by (2.3.28)and U jt by (2.4.2), we have a stronger estimate (2.2.29) for each ϕj , and summingthese inequalities gives

12td2(xk(t), v)− 1

2td2(x0(t), v) +

12t

k∑j=1

d2(xj(t), xj−1(t)) +k∑j=1

ϕj(xj(t)) 6 ϕ(v)

(2.4.12)

62

Section 2.4

for each t > 0 and v ∈ D(ϕ). Observe that inequality (2.4.11) is implied by theinequality (2.4.12), and in particular it holds when Ft is given by (2.3.28) and U jtby (2.4.2). Therefore in either case the inequality (2.4.9) and identity (2.4.8) canbe combined with (2.4.11), and after some basic algebra and a subtraction of apositive term at the right side of the combined inequality, we obtain the following:

12λd2(v, x0(t))− 1

2λd2(x, v) +

12λd2(x, x0(t)) +

k∑j=1

ϕj(xj(t)) 6 ϕ(v) (2.4.13)

holds for each v ∈ D(ϕ). Moreover if Ft is given by (2.3.28) and U jt by (2.4.2) weobtain the stronger inequality

12λd2(v, x0(t))− 1

2λd2(x, v) +

12λd2(x, x0(t))

+12t

k∑j=1

d2(xj(t), xj−1(t)) +k∑j=1

ϕj(xj(t)) 6 ϕ(v)(2.4.14)

for each v ∈ D(ϕ).

Proposition 2.4.2. For j = 0, 1, ..., k, xj(t)t6ε is a bounded subset of X forsome ε > 0. Moreover ϕj(xj(t))| t 6 ε, j = 1, 2, ..., k is a bounded subset of R.

Proof. We will derive both claims with the aid of inequalities (2.4.13) (for allv ∈ D(ϕ)), which hold in both cases. To this aim fix a z ∈ D(ϕ), and recall thecurves zj(t) = (U jt · · · U1

t )z from Lemma 2.4.1. The contraction property of theU jt ’s implies

d(xj(t), zj(t)) 6 d(xj−1(t), zj−1(t)) 6 · · · 6 d(x0(t), z) (2.4.15)

hencelim supt↓0

d(xj(t), zj(t))− d(x0(t), z) 6 0. (2.4.16)

Hence if we show that x0(t)t6ε is a bounded subset, then (2.4.3) and (2.4.15)guarantee that xj(t)t6ε is bounded too for j = 1, 2, ..., k. By (2.2.24) we canfind constants b, c ∈ R such that for all j’s and t > 0, ϕj(xj(t)) > c+ bd(x, xj(t)).Moreover the estimate

d2(x0(t), v) > (d(x0(t), x)− d(x, v))2

= d2(x0(t), x)− 2d(x0(t), x)d(x, v) + d2(x, v)

and (2.4.13) give (with c∗ := kc)

12λd2(x0(t), x)− 1

2λd(x0(t), x)d(x, v) + c∗ + b

k∑j=1

d(x, xj(t)) 6 ϕ(v) (2.4.17)

63



for each v ∈ D(ϕ). This inequality clearly implies that x0(t) is bounded if b > 0,so assuming that b < 0 does not reduce the level of generality in the first claim.Next for each j ∈ 1, ...k we have the following estimate:

d(x, xj(t)) 6d(x, U jt x) + d(U jt x, Ujt U

j−1t x) + d(U jt U

j−1t x, U jt U

j−1t U j−2

t x)

+ · · ·+ d(U jt · · ·U1t x, xj(t))

6j∑i=1

d(x, U itx) + d(x, x0(t)).

(2.4.18)We want to show that the quantities d(x, U itx) are bounded as t → 0 for eachi = 1, ..., k. Well if U jt = J jt for all j’s pick y ∈ D(|∂ϕj |) and observe that

d(x, U jt x) 6d(x, y) + d(J jt y, y) + d(J jt x,Jjt y)

62d(x, y) + d(J jt y, y),

which is bounded as t→ 0 by (2.2.32) and we used contraction property of the re-solvents (see Proposition 2.2.15). For the other choice of U jt the sequence d(x, U jt x)is bounded by continuity of gradient flow curves on [0,+∞) (see Theorem 2.2.20).Hence returning to (2.4.17), due to (2.4.18) we can conclude that

1λd2(x0(t), x)− αd(x0(t), x) 6 β for t small enough, (2.4.19)

for some constants α, β ∈ R depending only on v and x and not on t, whichguarantees that x0(t) is bounded as t → 0. As we already noticed, (2.4.16) thenimplies that xj(t)t6ε is also bounded and (2.2.24) implies that ϕj(xj(t))t6ε isbounded below for all j. Finally (2.4.13) implies ’a posteriori’ that ϕj(xj(t))t>εis bounded from above too, for all j under consideration.

Lemma 2.4.3. Let z ∈ D(ϕ). Then for j = 1, ..., k

d2(xj(t), z)− d2(x0(t), z) −→ 0 as t ↓ 0. (2.4.20)

Moreover

d(x0(t), xk(t)) −→ 0 as t ↓ 0. (2.4.21)

Proof. Let us show (2.4.21) first, which follows directly from the fact that x0(t)is bounded for small t > 0 (c.f. Proposition 2.4.2). Indeed as x0(t) = 1

1+λ/tx ⊕λ/t

1+λ/txk(t) we have d(x0(t), xk(t)) = tλd(x, x0(t))→ 0 as t ↓ 0.

64

Section 2.4

To prove (2.4.20) pick z ∈ D(ϕ) and 1 6 j 6 k. On one hand we have

d(xj(t), z)− d(x0(t), z) 6d(U jt z, z) + d(U jt Uj−1t z, U jt z)

+ d(U jt Uj−1t U j−2

t z, U jt Uj−1t z) + · · ·

+ d(U jt · · ·U1t z, xj(t))− d(x0(t), z)

6j∑i=1

d(U it z, z) + d(z, x0(t))− d(x0(t), z)

=j∑i=1

d(U it z, z)

(2.4.22)

and on the other hand by the contraction property of U jt we have

d(xj(t), z)− d(x0(t), z) > d(xk(t), Ukt · · ·Uj+1t z)− d(x0(t), z)

> d(xk(t), z)− d(z, Ukt · · ·Uj+1t z)− d(x0(t), z)

> −d(xk(t), x0(t))− [d(z, Ukt z)

+ d(Ukt z, Ukt U

k−1t z) + · · ·+ d(Ukt · · ·U

j+2t z, Ukt · · ·U

j+1t z)]

> −d(xk(t), x0(t))−k∑

i=j+1

d(z, U it z).

(2.4.23)

In light of Lemma 2.4.1 we see that limt↓0∑ki=1 d(U it z, z) = 0 for each z ∈ D(ϕ) ⊂

D(ϕi), i = 1, 2, · · · , k. Since

d2(xj(t), z)− d2(x0(t), z) = (d(xj(t), z)− d(x0(t), z)) (d(xj(t), z) + d(x0(t), z)) ,(2.4.24)

(2.4.21), (2.4.22), (2.4.23), and Proposition 2.4.2 yield (2.4.20).

Now we are able to prove our main results. It turns out that the Trotter-Katoproduct formula where the approximation steps are taken by the resolvent holds onCAT(0) spaces. We will also prove the version where the approximation steps aretaken by the semigroups under a suitable condition, which for example holds if theconsidered functionals are bounded from above on balls in X. (see Remark belowTheorem 2.4.5). As usually, in Theorem 2.4.4, Theorem 2.4.5, Corollary 2.4.7, andTheorem 2.4.8, J jt , t > 0 and (Sjt )t>0, denote the resolvents and the semigrouprespectively, associated to the functional ϕj for j = 1, 2, · · · , k, and (St)t>0 denotesthe semigroup associated to the functional ϕ defined in these theorems.

Theorem 2.4.4. Let X be a complete CAT(0) space, and let ϕ1, ..., ϕk : X →(−∞,+∞] be proper, l.s.c. convex functionals on X such that ϕ :=

∑kj=1 ϕj

is proper too. Then the following extension of the Trotter-Kato product formulaholds:(

J ktn J k−1

tn

· · · J 1tn

)nx −→ Stx as n→ +∞ for x ∈ D(ϕ) (2.4.25)

65



and the convergence is uniform on compact time intervals.

Proof. Recall the resolvents Jλ,t for λ > 0, t > 0 as constructed in Section 2.3,where Ft is given by (2.3.28). By Theorem 2.3.13 we only need to show thatfor x ∈ D(ϕ) and λ > 0, the convergence Jλ,tx → Jλx as t ↓ 0 holds. We fixsuch x and λ throughout the remainder of this proof. It is enough to show thatx0(tr) := Jλ,trx→ Jλx as r →∞ for an arbitrary sequence tr ↓ 0. Fix a sequencetr ↓ 0, and for j = 1, 2, · · · , k let (xj(tr))r be defined as in (2.4.7), where themappings U jt are defined by (2.4.2).

We will use the ultra-extensions ϕωj of ϕj for j = 1, 2, · · · , k from Section 2.2,relative to an arbitrary but fixed ultra-filter ω on N. Denote for j = 0, 1, · · · , kthe ultra-limit xj := [(xj(tr))] ∈ Xω. This is well defined by Proposition 2.4.2,and note moreover that by the same proposition xj ∈ D(ϕωj ) for j = 0, 1, · · · , k.Next by (2.4.14) we have for v ∈ D(ϕ) 6= ∅

2trϕ(v) > d2(xk(tr), v)− d2(x0(tr), v)

+k∑j=1

d2(xj(tr), xj−1(tr)) + 2trk∑j=1

ϕj(xj(tr))(2.4.26)

and Proposition 2.4.2 and (2.4.20) now yield that

lim supr→∞

n∑j=1

d2(xj(tr), xj−1(tr)) 6 0 (2.4.27)

hence limr→ω d(xj(tr), xj−1(tr)) = 0 holds for j = 1, 2, · · · , k. In particular,

x0 = x1 = · · · = xn. (2.4.28)

In what follows we will abuse the notation slightly by writing y instead of [(yn)],yn := y, ∀n ∈ N, for y ∈ X. Let vj ∈ D(ϕωj ) for j = 1, 2, · · · , k. Let moreover(vjn) ⊂ D(ϕj) be a bounded sequence such that limn→ω v

jn = vj for j = 1, 2, · · · , k.

Then the inequality (2.4.10) with v := vjn holds for each j ∈ 1, 2, · · · , k andfor each n ∈ N. Then taking the ultra-limit of both sides of these inequalities,and subsequently taking the infimum at the right hand side over all sequences(vjn)n ∈ vj for j = 1, 2, · · · , k, we conclude that

12tr

d2ω(xj(tr), v)− 1

2trd2ω(xj−1(tr), v) + ϕωj (xj(tr)) 6 ϕωj (v) (2.4.29)

holds for each v ∈ D(ϕωj ), j = 1, 2, · · · , k and r ∈ N. Define the functionalψω :=

∑kj=1 ϕ

ωj : Xω → Xω. Observe that ψω is convex on Xω since it is a sum

of convex functionals by Lemma 2.2.29, and observe moreover that ψω ≡ ϕ holdson X and D(ϕ) ⊂ D(ψω).7

7Clearly ψω 6 ϕω holds here, but it is not clear to the author whether ψω = ϕω holds, hencea modification of the straightforward argument with the aid of ϕω is necessary.

66

Section 2.4

Summing inequalities (2.4.29) for j = 1, 2, · · · , k, we obtain

12tr

d2ω(xn(tr), v)− 1

2trd2ω(x0(tr), v) +

n∑j=1

ϕωj (xj(tr)) 6 ψω(v) (2.4.30)

for each v ∈ D(ψω) ⊃ D(ϕ) 6= ∅. Since X ⊂ Xω and Xω is a complete CAT(0)space too, the curves s 7→ (1 − s)x ⊕ sxn(tr) are geodesics in Xω for each r ∈ Nand we can do the same estimate as we did to obtain (2.4.13) but this time witha base point v ∈ D(ψω). What we get is the following inequality:

12λd2ω(x, x0(tr))−

12λd2ω(x, v) +

n∑j=1

ϕj(xj(tr)) 6 ψω(v) (2.4.31)

for each v ∈ D(ψω). Since d2ω(x, x0(tr)) = d2(x, x0(tr)) for all r ∈ N we can take

the ultra-limit for r → ω in (2.4.31). Recalling the definition of the ultra-extensionof the ϕj ’s we conclude that

12λd2ω(x, x0)− 1

2λd2ω(x, v) + ψω(x0) 6 ψω(v) (2.4.32)

for v ∈ D(ψω). Now as the sequence tr ↓ 0 was arbitrary the same conclusion(2.4.32) holds for any subsequence of tr, i.e. it holds for any subsequence ofx0(tr)r. Therefore Lemma 2.2.30 implies that x0 ∈ X and that x0(tr) convergesto x0 in X, provided that we show that the functional w 7→ 1

2λd2ω(x, w) + ψω(w)

admits at most one minimum point in Xω. This uniqueness follows by convexity ofψω and strict convexity of w 7→ 1

2λd2ω(x, w): if there were two minimizers w0 6= w1

then their middle point z := 12 w0 ⊕ 1

2 w1 would satisfy

12λd2ω(x, z) + ψω(z) 6

12

12λd2ω(x, w0) +

12

12λd2ω(x, w1)− 1

212

12λd2ω(w0, w1)

+12

12λψω(w0) +

12

12λψω(w1) <

12λd2ω(x, w0) + ψω(w0)

which is a contradiction. Since inequality (2.4.32) in particular holds for eachv ∈ D(ϕ) ⊂ D(ψω), and ϕ(x) = ψω(x) for each x ∈ D(ϕ), and d = dω on X ×X,we conclude that x0 = Jλx and the proof is complete.

Theorem 2.4.5. Let X be a complete CAT(0) space, and let ϕ1, ..., ϕk : X →(−∞,+∞] be proper, lower semi-continuous convex functionals such that ϕ :=∑nj=1 ϕj is proper. Assume that for j = 1, ..., n the functional ϕj has the property

that D(ϕωj )Xω

=(Dj

)ω. Then the following extension of the Trotter-Kato formula

holds: (Sktn Pk Sk−1

tn

Pk−1 · · ·S1tn P1

)n−→ Stx as n→∞ (2.4.33)

for x ∈ D(ϕ) and the convergence is uniform on compact time intervals. Inparticular this formula holds if ϕ1, ..., ϕk−1 are bounded from above on each ball inX.

67



Remark 2.4.6. Since Dj ⊂ X is a metric space, one can consider its ultra-limitDj

ω(which is a complete CAT(0) space by Theorem 2.2.27, since Dj being a closed

convex subset of a CAT(0) space, is also a complete CAT(0) space). Moreover onecan consider the subspace Ejω := x ∈ Xω| ∃ (xn)n ∈ x, xn ∈ Ej ∀n ∈ N of Xω.Clearly Dj

ωand Ejω are isometric spaces, therefore we will regard Dj

ωto be a

subspace of Xω. It is easy to see that D(ϕωj )Xω

⊂(D(ϕj)

)ω, but it is not clear to

the author whether the opposite inclusion holds in general. However, a reasonablesituation when it does hold, is when ϕj is bounded above on each bounded subsetof X.

Proof. (of Theorem 2.4.5)As in the proof of Theorem 2.4.4, we will argue with the aid of Theorem 2.3.13:

it suffices to show that for each x ∈ D(ϕ) and λ > 0 the convergence Jλ,trx→ Jλxas r → +∞ holds for arbitrary sequence tr ↓ 0 (approximate resolvents Jλ,t areconstructed in Section 2.3, and the mappings Ft and U jt are defined by (2.3.27) and(2.4.1) respectively), and we fix such x and λ. Recall moreover (2.4.6) and (2.4.7),and denote the ultra-limit xj := [(xj(tr))] for j = 0, 1, , · · · , k, which is well defineddue to Proposition 2.4.2. Moreover the same proposition implies that xj ∈ D(ϕωj )for j = 1, 2, · · · , k. Now observe that if we establish that x0 = x1 = · · · = xk

holds, than we can repeat the arguments given below display (2.4.28) in the proofof Theorem 2.4.4, and conclude convergence (2.4.33). Indeed these arguments aredue to inequality (2.4.10), which holds in the setting of this theorem as well.

Let us prove that x0 = x1 = · · · = xk holds. By (2.2.39) we have for eachk ∈ N and j = 1, 2, · · · , k

d2(xj(tr), v)− d2(Pjxj−1(tr), v) + 2trϕj(xj(tr)) 6 2trϕj(v)

for all v ∈ D(ϕj) 6= ∅. Hence by Proposition 2.4.2

lim supr→∞

(d2(xj(tr), v)− d2(Pjxj−1(tr), v)

)6 0, ∀v ∈ D(ϕj), (2.4.34)

so that we also haved2ω(xj , v)− d2

ω(zj , v) 6 0 (2.4.35)

for each v ∈ D(ϕωj ) where we denote xj := [(xj(tr))] and zj = [(Pjxj−1(tr))] ∈ Xω.On the other hand by (2.2.18) for any v ∈ D(ϕj) 6= ∅

d2(xj−1(tr), v)− d2(Pjxj−1(tr), v) > d2(xj−1(tr), Pjxj−1(tr)) (2.4.36)

Now (2.4.34) and Lemma 2.4.3 together imply that

lim supr→∞

(d2(xj−1(tr), v)− d2(Pjxj−1(tr), v)

)6 0 ∀v ∈ D(ϕj) 6= ∅ (2.4.37)

and this together with (2.4.36) yields

d2(xj−1(tr), Pjxj−1(tr)) −→ 0 as r →∞ (2.4.38)

68

Section 2.4

and in particular xj−1 = zj . Now as zj ∈ (Dj)ω by definition, due to the assump-

tion D(ϕωj ) =(D(ϕj)

)ω, we may take v = zj = xj−1 in (2.4.35), which yields

that xj = xj−1 for all j. The proof is now complete.

For a finite set N , the number of elements in N is denoted by #N .

Corollary 2.4.7. Let X be a complete CAT(0) space, and let ϕ1, ϕ2, · · · , ϕk beproper, lower semi-continuous, geodesically convex functionals defined on X, suchthat ϕ :=

∑kj=1 ϕj is proper. For j = 1, 2, · · · , k let U jt be given either by (2.4.1),

or by (2.4.2), for all t > 0. If k = 2 then(J 2tnS1tn

)n,(S2tnJ 1tn

)nx −→ Stx, n→ +∞ as n→ +∞ (2.4.39)

holds for each x ∈ D(ϕ), and the convergence is uniform on compact time intervals.Denote N ⊂ 1, 2, · · · , k to be the set of j’s such that U jt is given by (2.4.1),

and assume that there are #N − 1 indexes j ∈ N , such that the functional ϕjsatisfies D(ϕωj )

Xω

=(Dj

)ω. Then the following mixed version of the Trotter-Kato

product formula holds:(Uktn Uk−1

tn

· · · U1tn

)x −→ Stx as n→ +∞, (2.4.40)

for each x ∈ D(ϕ), and the convergence is uniform on compact time intervals. Inparticular, (2.4.40) holds if for #N −1 indexes j ∈ N , we have that ϕj is boundedfrom above on each ball in X.

Proof. Let us prove (2.4.40) first. In light of Theorem 2.3.13, it is enough to showthe convergence of resolvents Jλ,tx → Jλx as t ↓ 0 holds, for each x ∈ D(ϕ) andλ > 0. We use the same notation as in the proofs of Theorem 2.4.4 and Theorem2.4.5, and we fix x ∈ D(ϕ), λ > 0, and a sequence tr ↓ 0. It is enough to showthat Jλ,trx → Jλx as r → +∞. Define xj(tr) for j = 0, 1, · · · , k and r ∈ N.Inspect the proof of Theorem 2.4.4 carefully, to conclude that xj := limr→ω xj(tr)is well defined for j = 0, 1, · · · , k. Moreover, we have that xj = xj−1 for j ∈1, 2, · · · , k \ N . Inspecting the proof of Theorem 2.4.5 carefully, we conclude

that if D(ϕωj )Xω

=(Dj

)ω, then we have that xj = xj−1. Thus by the assumption

of the corollary, xj = xj−1 holds for each j ∈ N . Notice moreover, that by Lemma2.4.3 xk = x0. Hence there are k different j’s in 1, 2, · · · , k such that xj = xj−1,which implies that x0 = x1 = · · · = xk holds. Since (2.4.10) holds in the settingof this corollary, we can repeat the exposition given in the proof of Theorem 2.4.4below (2.4.28), and conclude (2.4.40). The version of the Trotter-Kato productformula (2.4.39) now follows since 1 + 1 = 2.

There is another instance when the product formula holds.

Theorem 2.4.8. Let X be a complete CAT(0) space, and let ϕ1, ..., ϕk : X →(−∞,+∞] be proper, l.s.c. and convex and let ϕ :=

∑kj=1 ϕj 6= +∞. For j =

69



1, 2, · · · , k, let U jt be defined by (2.4.1) or by (2.4.2) for each t > 0. Assumemoreover that there is a j0 ∈ 1, ..., k such that all bounded subsets of each levelset of ϕj0 is relatively compact in X. Then the following product formula holds:

(Uktn · · · U1

tn

)nx −→ Stx as n→∞, ∀t > 0, , ∀x ∈ D(ϕ), (2.4.41)

the convergence being uniform on compact time intervals.

Proof. In light of Theorem 2.3.13 we only need to show that the convergence ofapproximate resolvents x0(t) = Jλ,tx→ Jλx as t ↓ 0 holds for each x ∈ D(ϕ) andeach λ > 0. Let j0 ∈ 1, ..., k be such that ϕj0 has the stated property. Then as(xj0(t))t>0 is bounded by Proposition 2.4.2, for each sequence tr ↓ 0, the sequence(xj0(tr))r has a convergent subsequence (xj0(trl))rl → xλ ∈ X. By Lemma 2.4.3the sequence (xj(trl))rl must converge to xλ as well, for all j ∈ 1...., k. Nowrecall (2.4.13) and estimate

12λd2(v, xλ)− 1

2λd2(x, v) +

12λd2(x, xλ) +

k∑j=1

ϕj(xλ)

6 liml→+∞

(1

2λd2(v, x0(trl))−

12λd2(x, v) +

12λd2(x, x0(t))

)+

k∑j=1

ϕj(xλ)

6 liml→+∞

(1

2λd2(v, x0(trl))−

12λd2(x, v) +

12λd2(x, x0(t))

)+

k∑j=1

lim infl→+∞

ϕj(xλ)

6 lim infl→+∞

12λd2(v, x0(trl))−

12λd2(x, v) +

12λd2(x, x0(trl)) +

k∑j=1

ϕj(xj(trl))

6ϕ(v), ∀ v ∈ D(ϕ)

with the aid of the lower semi-continuity of ϕj ’s. Hence xλ = Jλx, and since thesequence tr ↓ 0 was arbitrary, we conclude that the convergence of the approximateresolvents holds. We have proved the theorem.

Remark 2.4.9. Assume the setting of Theorem 2.4.8, and suppose that X islocally compact. Then due to the Hopf-Rinow Theorem 2.2.12, any functionalsatisfies the level sets compactness assumption of Theorem 2.4.8. Consequently(2.4.41) always holds in locally compact, complete CAT(0) spaces.

70

Section 2.5

2.5 Examples

Clearly, our Theorem 2.4.4 generalizes the classical i.e. Hilbert space situation. Awell known example within this theory is the heat equation functional

ϕ(u) :=∫

Ω

|∇u|2dx if u ∈W 1,2(Ω),

ϕ(u) := +∞ if u ∈ L2(Ω) \W 1,2(Ω),(2.5.1)

where Ω is a suitable domain in Rk. Then ϕ :=∑kj=1 ϕj where ϕj(u) =

∫(∂ju)2dx

are defined on a suitable subset of L2(Ω), for j = 1, 2, · · · , k. These functionalsare convex, proper and lower semi-continuous and as L2(Ω) is complete CAT(0)space our theory can be applied.

In the general setting of a CAT(0) space X without any additional structure,the most natural convex and lower semi-continuous functionals are

ϕ(x) :=12d2(x, z) x ∈ X (2.5.2)

where z ∈ X is a fixed point. Moreover one can consider any positive linearcombination of such functionals. So let us fix distinct points z1, ..., zn in a CAT(0)space X, and positive numbers α1, ..., αn > 0, and define functionals

ϕj(x) :=αj2d2(x, zj), j = 1, ...k , x ∈ X, (2.5.3)

ϕ(x) :=k∑j=1

αj2d2(x, zj), x ∈ X. (2.5.4)

Each of these functionals is clearly continuous, and the CAT(0) condition is clearlyeven stronger then the convexity along geodesics of these functionals. Moreoversuch functionals are clearly bounded on balls, so any of the product formulas givenin Section 2.4 holds. Furthermore one can consider the functionals x 7→ dp(x, zj)where p > 1. Such functionals are convex, continuous, and bounded on balls,hence any version of the product formula holds here as well.

A more complex example is when one considers a gradient flow associated to aconvex l.s.c. functional ϕ on CAT(0) space X but wants to constrain it to a closedconvex subset C ⊂ X. In this context convexity is understood in the sense of theCAT(0) metric of our space. For example if X is a Hilbert ball, such convex subsetmay not be convex in the usual linear sense. Or we can consider a gradient flow onthe space of positive unitized Hilbert-Schmidt operators equipped with the traceinner product (c.f. [62]), which induces a different geometry from the one inducedby the operator norm. In order to construct such a flow one can add the indicatorfunctional

ϕC(x) := 1 if x ∈ C, ϕC(x) := +∞ if x ∈ X \ C (2.5.5)

to ϕ , i.e. consider the flow associated to ϕ1 := ϕ+ϕC . Then it can be very hardto get any idea about the paths of the constrained flow. However, the resolvents

71



associated to the indicator functional ϕC are easily seen to be the nearest pointprojections onto C, and its semigroup is just the identity map on C. Thereforeapplying the product formula provides a possibility to gain insight in the con-strained equation. We refer to [63] and [66] for some concrete examples in thelinear setting.

72

Chapter 3

Wasserstein-2 analysis of thenon-symmetricFokker-Planck equation andthe Trotter-Kato ProductFormula

This chapter has been submitted for publication as: I. Stojkovic, “Wasserstein-2analysis of the non-symmetric Fokker-Planck equation and the Trotter-Kato Prod-uct Formula”.

In this chapter we study several aspects of the non-symmetric Fokker-Planck equa-tion as a flow on the Wasserstein-2 space. The first main novelty of our research isa Trotter-Kato like product formula and we will show that the limit in this productformula gives solutions of the non-symmetric Fokker-Planck equation and inducesa contraction semigroup on the Wasserstein-2 space (P2(Rd),W2) of probabilitymeasures on Rd. Our product formula is also a new result in the special case wherethe drift of the equation is a gradient. However, if the drift is not a gradient thenthere is no reason why this equation would correspond to a gradient flow in thesense of Ambrosio-Gigli-Savare, therefore the standard theory of gradient flows onP2(Rd) can not be applied in the non-symmetric case. Another novelty of this re-search, is our proof of the regularising effect for the non-symmetric Fokker-Planckequation, and we will argue this proof with the aid of our product formula. Wewill conclude a theorem stating properties of the semigroup of solutions, which areanalogous to the properties of the flow induced by a maximal monotone operatoron a Hilbert space.

73

CHAPTER 3: Wasserstein-2 analysis of the non-symmetric Fokker-Planck equation and

the Trotter-Kato Product Formula

3.1 Introduction

The non-symmetric Fokker-Planck equation on Rd is the following evolution equa-tion:

∂tµt = ∆µt +∇ · (bµt) µ0 ∈ P(Rd) (3.1.1)

in the sense of distributions on D((0,∞)×Rd), where the solution t 7→ µt assumesvalues in the space P(Rd) of Borel probability measures on Rd and where the driftcoefficient b : Rd → Rd is assumed to be monotone, i.e. 〈b(x)− b(y), x− y〉 > 0 forall x, y ∈ Rd. More precisely, a curve 0 6 t 7→ µt ∈ P(Rd) is a solution of (3.1.1)in the sense of distributions, if∫ ∞

0

∫Rd

(∂tψ(t, x)−∆xψ(t, x) + 〈∇ψx(t, x), b(x)〉) dµt(x) dt = 0,

∀ψ ∈ C∞c ((0,+∞)× Rd).(3.1.2)

We use the standard notation, and denote D(Ω) = C∞c (Ω) to be the linear spaceof compactly supported real valued infinitely often differentiable functions definedon an open subset Ω ⊂ Rn, for n ∈ N.

If b is not a gradient, then equation (3.1.1) is not a gradient flow equationon P2(Rd), in the sense of Ambrosio-Gigli-Savare. Therefore, the equation (3.1.1)does not fall within the scope of the theory presented in [5] or [106].

In this chapter, we study the Fokker-Planck equation (3.1.1) as a flow onthe Wasserstein-2 space P2(Rd). The main results of our investigations are thefollowing. Firstly, we prove the Trotter-Kato product formula for the Fokker-Planck equation (3.1.1), that is

(R2t/nR

1t/n)nµ W2−→ Rtµ for t > 0, µ ∈ P2(Rd), (3.1.3)

where (R1t )t>0 denotes the transport semigroup induced by b, (R2

t )t>0 denotes theheat semigroup, and (Rt)t>0 denotes the semigroup induced by the non-symmetricFokker-Planck equation (3.1.1). Secondly, we show that the semigroup (Rt)t>0

possesses the regularising effect, similar to the semigroups induced by the gradientflow equations.

The existing theory about gradient flows on the Wasserstein-2 space does notyield any claim about equation (3.1.1) directly. For if b is not a gradient of afunction on Rd, there is no reason why solutions of (3.1.1) should be gradient flowcurves on (P2(Rd),W2) in the sense of Ambrosio-Gigli-Savare [5] (see also [7] for thestudy of the symmetric Fokker-Planck equation in the infinite dimensional setting).A proof of the claim that there is no convex functional ϕ : P2(Rd) → (−∞,+∞]which induces solutions of (3.1.1) would be much less surprising than a proof thatthere is such convex functional.

One of the most striking developments in the theory of PDE’s of the pastdecade is the treatments of various classes of PDE’s as gradient flow evolutions onthe Wasserstein-2 space P2(Rd) of Borel probability measures on Rd with finitesecond moments. This novel branch of variational analysis has been initiated by

74

Section 3.1

seminal papers [54] by Jordan, Kinderlehrer and Otto, and [85] by Otto, wherethe symmetric Fokker-Planck equation (i.e. equation (3.1.1) with b := ∇V for aconvex function V : Rd → R), and the porous medium equation, respectively, aretreated in such a way. Subsequently many other authors have conducted theirown investigations (among numerous other sources we refer to [5], [22], [107] and[106]). This exciting new branch of variational analysis has evolved to a rathermature theory, and it is usually referred to as the Optimal Transportation theory.

Even if the PDE in question is already known to have unique solutions withina certain class of measures by a traditional analytical method, showing that it is agradient flow on P2(Rd) provides a very interesting interpretation for the equation.This interpretation also provides the Euler backward numerical approximation, i.e.the resolvent operators, the optimal order of convergence of these approximationsto the solution and the contraction property of the semigroup of solutions withrespect to the natural W2 distance. Moreover such a semigroup exhibits the reg-ularising effect which implies strong path regularity properties. If the PDE underconsideration has an invariant state, we may also have a strong estimate of conver-gence to an invariant state with respect to the Wasserstein-2 distance. A classicaltreatment of such types of equations does not always produce such results, andmany authors have found it interesting to study such equations in the new way. Adetailed comparison of both mathematics and physics of the classical versus theWasserstein approach of various classes of equations can be found in [85] and [106].

Second order linear PDE’s and the corresponding SDE’s have been studiedextensively in the past, and even more recently, several papers have appearedabout this topic (see [14] for existence, [13] for uniqueness and [12] for the regularityunder conditions allowing for rather singular coefficients). Contrary to the authors’aspiration in the above mentioned monographs to extend the level of generalityof the coefficients appearing in the equation, the object of our study is to showthat the solutions posses properties which are similar to the properties that thegradient flows have. A similar approach has been taken in [80] where the authorsinvestigate (3.1.1) and prove the contraction property of the flow of solutionswith respect to a class of optimal transportation distances which contain all theWasserstein-p distances for p ∈ [1,∞). Further, in the final chapter of his recentPhD thesis [79] L.Natile constructs solutions if the initial condition is an absolutelycontinuous measure which is moreover in the domain of the metric slope of therelative entropy functional of the d−dimensional Lebesgue measure and he alsoshows that they posses some additional properties.

One of the main novelties of our investigations is the Trotter-Kato productformula for (3.1.1). In light of the continuity equation on (P2(Rd),W2) (see [5]Chapter 8) the Wasserstein ’generator’ of (3.1.1) can be seen as a sum of thesubdifferential of the Heat Entropy functional and the mapping

B(µ) := Pµb, µ ∈ P2(Rd), (3.1.4)

where Pµ : L2(µ; Rd) → TanµP2(Rd) denotes the orthogonal projection onto theregular tangent space TanµP2(Rd) of P2(Rd) at µ. Moreover, both of these ’gen-erators’ induce a semigroup which has nice geometric properties. We will exploit

75



this fact to construct the semigroup on P(Rd) of solutions of (3.1.1) as the limitof the successive iterations of small steps along the paths of these two semigroups.

For the verification of one technical step in establishing the product formula, weassume b to be Lipschitz continuous. For all other steps the weaker at most lineargrowth condition (together with the maximum monotonicity property) suffices (seeRemark 3.3.6).

Even if b is a gradient of a convex function V : Rd → R, our product for-mula is a new result. In this case (3.1.1) is called the symmetric Fokker-Planckequation. This case is quite different from the general case, as then (3.1.1) is agradient flow equation. A different version of the product formula for the symmet-ric Fokker-Planck equation (i.e. one assumes that b = ∇V for a convex functionV : Rd → R which is a gradient flow equation) with both steps in the Trotter-Katoapproximations being obtained by the resolvents rather than by the semigroupshas also been considered under more restrictive conditions in [27]. Clearly evenif we assume that b is a gradient, our product formula is a new result and of agreater level of generality. Since the transport semigroup as well as the heat equa-tion semigroup are contractions with respect to the Wasserstein-2 distance W2 ourproduct formula directly recovers the W2-contraction property proved in [80].

We also show new regularising properties. The semigroup of the solutions of(3.1.1) that we construct exhibits a regularising effect (for precise formulation seeSection 3.4) and locally Lipschitz path properties, which we prove with the aidof the product formula. If the semigroup of solutions of an equation does notposses such a regularising effect then paths which start in the boundary of thedomain of the ’generator’ may not be solutions of the equation. Therefore ourresult yields an improvement with respect to the existence result in Chapter 5of [79], since there the initial measure is required to be an absolutely continuousmeasure and moreover a member of the domain of the metric slope of the relativeentropy functional.

In light of the analogy between the theory of gradient flows on P2(Rd) versusgradient flows on Hilbert spaces (see [5] and [106]), and due to the fact that the flowinduced by the symmetric Fokker-Planck equation is a gradient flow on P2(Rd)(see [54] and [5]), it seems that the geometric structure of the flow on P2(Rd)induced by (3.1.1) can be compared with the structure of the flow induced bya maximal monotone operator on a Hilbert space. Therefore the results of thischapter indeed confirm that the Wasserstein-2 space is a natural ambient space tostudy the non-symmetric Fokker-Planck equation.

A crucial assumption in the analysis that we are going to present here is thefact that B is monotone along any coupling plan of any pair of measures. Wewill prove the product formula with the aid of the Markov kernels of the SDEassociated to 3.1.1 (see (3.2.7)). In [109] and [23] the authors consider stochasticinclusions instead of an SDE, with a multi-valued maximal monotone operator asa drift. It may be expected that a generalisation of our results in this directioncan be argued for in a similar fashion.

76

Section 3.2

3.2 Preliminaries

In this section we will recall some facts which we need for our analysis. In thefirst part of this section we will recall the stochastic differential equation (SDE)associated to the equation (3.1.1), its backward equation and the Markov kernelsand their relation with the Fokker-Planck equation. In the second part of thissection we recall the Wasserstein-2 theory of the Heat equation.

The drift coefficient b : Rd → Rd appearing in (3.1.1) will be assumed to satisfythe following two conditions.

1. b is monotone, i.e.

〈b(x)− b(y), x− y〉 > 0 ∀x, y ∈ Rd, (3.2.1)

2. b is Lipschitz, i.e.

∃C > 0 : |b(x)− b(y)| 6 C|x− y| ∀x, y ∈ Rd. (3.2.2)

If the second condition above holds, then b also satisfies the linear growthcondition

∃C > 0 : |b(x)| 6 C(1 + |x|) ∀x ∈ Rd (3.2.3)

and although we assume the stronger Lipschitz condition in order to prove theproduct formula in Section 3.3, the preliminary results in Section 3.3 are provenunder the weaker assumption (3.2.3). We will comment in detail about this inSection 3.3. The term ’non-symmetric’ originates from the case where b := A ∈ Rd2

is a linear mapping, for in such case A is a gradient if and only if it is a symmetricmatrix since we always have ∇(x 7→ 1

2 〈Ax, x〉) = A+A∗

2 .Next let us recall some facts about maximal monotone subsets A ⊂ Rd ×

Rd. Such a subset A is said to be a λ-monotone operator, for λ ∈ R, if for any(x1, y2), (x2, y2) ∈ A we have that

〈x1 − x2, y1 − y2〉 > λ|x1 − x2|2. (3.2.4)

If λ = 0 we will omit the prefix λ and call A monotone. A is called maximalmonotone if there is no monotone subset B ⊂ Rd × Rd that contains A properly.Notice that Amay be multi-valued and if it is single valued and defined everywhere,then we will say that A is a mapping and write A : Rd → Rd.

Next a single valued maximal monotone mapping b : Rd 7→ Rd is always contin-uous by [3] Corollary 1.3 (4). On the other hand any continuous monotone (singlevalued) mapping defined on Rd is maximal monotone by [17] Proposition 2.4. Letus gather these facts in the following lemma.

Lemma 3.2.1. Let A : Rd → Rd be a monotone mapping. Then A is maximal ifand only if it is continuous.

Any maximal monotone subset b of Rd with dense domain in Rd induces acontraction semigroup (St)t>0 on Rd. In such case each path t 7→ Stx of theinduced semigroup is Lipschitz on [0,∞) for each x ∈ Rd and the equation

ddtStx = −b(Stx), L1-a.e t ∈ [0,∞), ∀x ∈ Rd (3.2.5)

77



holds (see [17] Theorem 3.1). Moreover the function

t 7→ |b(Stx)| (3.2.6)

is non-increasing for each x ∈ Rd. Throughout the remainder of this chapter, wefix a single valued maximal monotone mapping b : Rd → Rd . Moreover (St)t>0

denotes the induced semigroup associated to b, i.e. the semigroup whose pathsgive the unique solutions of (3.2.5).

Equation (3.1.1) is the Kolmogorov forward equation of the associated stochas-tic differential equation (SDE)

dXt = −b(Xt) dt+√

2 dWt (3.2.7)

where (Wt)t>0 is a standard Brownian motion with values in Rd. Equations of thistype have alomst surely unique solutions for any initial condition, provided (3.2.1)and (3.2.3) are satisfied (see [61] Chapter V Theorem 1 where more general con-ditions are stated). Moreover under the same assumptions (3.2.1) and (3.2.3) dueto weak uniqueness (see [39] Chapter 8 Theorem 2.6) the equation (3.2.7) inducesa Markov family with corresponding Markov transition kernels. The Kolmogorovbackward equation associated to (3.2.7) is the following equation

∂tρt = ∆ρt − 〈∇ρt, b〉 (3.2.8)

and its formal adjoint equation is the Fokker-Planck equation (3.1.1). Thus thepaths of the dual semigroup (the semigroup induced by the action of the transitionkernels on measures), or said differently the curves of laws of solutions of (3.2.7)are the solutions of (3.1.1) for any initial µ0 ∈ P(Rd). This claim follows by adirect application of the Ito formula (general theory of SDE’s and Markov processescan be found in [39], [98], [94] among other sources, while [32] exposes theory oninfinite dimensional spaces).

For x ∈ Rd we will denote (Xxt )t>0 to be the up to indistinguishability unique

solution of (3.2.7) which starts in x a.s. Since Brownian motion is a Levy process,with the aid of the weak uniqueness of the solutions of (3.2.7), one can show thatthese solutions form a Markov family. We denote (Qt)t>0 to be the associatedMarkov semigroup which is given by

Qtf(x) = Ef(Xxt ), f ∈ bB(Rd), t > 0. (3.2.9)

By (Rt)t>0 we define the restriction of the dual semigroup of (Qt)t> to P(Rd),which is defined by

Rtµ(B) := Q∗tµ(B) =∫

RdQt1B dµ(x) µ ∈ P(Rd), B ∈ B(Rd), t > 0. (3.2.10)

As usually bB(Rd) denotes the linear space of bounded real valued Borel Mea-surable functions defined on Rd and P(Rd) denotes the set of Borel probabilitymeasures on Rd. Due to a.s. continuity of the solutions of (3.2.7) one immediately

78

Section 3.2

sees that for any µ ∈ P(Rd) the curve 0 6 t 7→ Rtµ is weakly continuous. Noticethat an easy application of the Ito formula indeed yields that the paths of (Qt)t>0

are solutions of equation (3.2.8), since

ddt

(Qtf) = ∆(Qtf)− 〈∇(Qtf), b〉, t > 0, (3.2.11)

for sufficiently smooth initial condition f at least in a pointwise sense. We willdenote µt := Law(Xt) the distribution of the solution at time t > 0 if it is clearfrom the context what the initial distribution is.

Furthermore we consider the following two semigroups and their dual semi-groups. As usually C0(Rd) denotes the Banach space of continuous real valuedfunctions defined on Rd which vanish at infinity, equipped with the supremumnorm, and C∞c (Rd) denotes its subspace of infinitely differentiable compactly sup-ported functions. We define

P 1t f(x) := f(Stx), t > 0, x ∈ Rd, f ∈ bB(Rd), (3.2.12)

P 2t f(x) :=

1√(2π)dtd

∫f(y)e−|x−y|

2/2t dy, t > 0, x ∈ Rd, f ∈ bB(Rd),

P 20 f(x) :=f(x), x ∈ Rd, f ∈ bB(Rd),

(3.2.13)R1tµ := (P 1

t )∗µ, µ ∈ P(Rd), (3.2.14)

R2tµ := (P 2

t )∗µ, µ ∈ P(Rd). (3.2.15)

It is very well known that C0(Rd) is invariant under the action of the semigroup(P 2t )t>0 (and it is also not hard to show). We will argue that under the conditions

(3.2.1) and (3.2.3) the function space C0(Rd) is also invariant under the action ofsemigroups (P 1

t )t>0 and (Qt)t>0 and moreover that their restrictions to C0(Rd)are C0-semigroups. We will prove these claims in the next section. Observe thatwe have

R2tµ(B) =

1√(2π)dtd

∫B

e−|x−y|2/2t dµ(x) (3.2.16)

for B ∈ B(Rd), µ ∈ P(Rd) and t > 0.Once we have established the above mentioned facts about the semigroups

(P 1t )t>0 and (Qt)t>0 we will use these facts to show that the Trotter-Kato product

formula (3.1.3), holds (the Wasserstein-2 distance W2 is defined in (3.2.20) below).Notice that (3.1.3) directly implies that (Rt)t>0 is a W2-contraction semigroup.This result will turn out to be very useful in analysing the path properties of(Rt)t>0 (see section 4).

Let us recall some basic facts about the Wasserstein-2 space of Borel probabilitymeasures on Rd. In the sequel P2(Rd) will denote the set of probability measuresµ on Rd such that

∫|x|2 dµ(x) < +∞. For µ ∈ P2(Rd) the set of Borel maps

79



g : Rd → Rd such that∫

Rd |g(x)|2 dµ(x) < +∞ is as usually denoted by L2(µ; Rd).The projection maps are defined by

π1, π2 : R2d → Rd π1(x, y) := x, π2(x, y) := y, x, y ∈ Rd, (3.2.17)

and for a measure γ ∈ P2(R2d) we define

π1#γ(B) := γ(B × Rd), π2

#γ(B) := γ(Rd ×B) for B ∈ B(Rd). (3.2.18)

For a pair of measures µ1, µ2 ∈ P2(Rd) the set of transport plans is defined by

Γ(µ1, µ2) := γ ∈ P2(Rd) : πj#γ = µj for j = 1, 2. (3.2.19)

One can define a metric on P2(Rd) by

W 22 (µ1, µ2) := inf

γ∈Γ(µ1,µ2)

∫R2d|x− y|2 dγ(x, y), µ1, µ2 ∈ P2(Rd), (3.2.20)

and the set of minimizers in (3.2.20), i.e. the set of ’optimal transport plans’denoted Γo(µ1, µ2) is always non-empty. One can show that (P2(Rd),W2) is acomplete separable metric space (see [5] Remark 7.1.7). Denoting δ0 to be theDirac measure at 0, we have the following expression:

W 22 (µ, δ0) =

∫Rd|x|2 dµ(x) ∀µ ∈ P2(Rd). (3.2.21)

If µ1 Hd−1 where Hd−1 denotes the (d− 1)-dimensional Hausdorff measureon Rd, and in particular if µ1 Ld (here and in the sequel Ld denotes theLebesgue measure on Rd), then Γo(µ1, µ2) is a one point set and there is a mappingr ∈ L2(µ1; Rd) such that the unique optimal transport plan is concentrated on thegraph of r. We then have that r#µ1 = µ2, and also

W 22 (µ1, µ2) =

∫|x− r(x)|2 dµ1(x). (3.2.22)

Moreover, in that case there is a convex function ψ on Rd such that r = ∇ψ forµ1 a.e. x ∈ Rd and such a map r is called the optimal transport map. For proofsof these claims we refer to [42]. One can also define spaces Pp(Rd) for 1 6 p < +∞by replacing the constant 2 in (3.2.20) by p. A subset C ⊂ Pp(Rd) has uniformlyintegrable p-th moments if

supµ∈C

∫|x|>R

|x|p dµ(x)→ 0 if R→ +∞. (3.2.23)

Moreover, such a subset C is relatively compact in Pp(Rd) if and only if its p-thmoments are uniformly integrable. Therefore it is not hard to see that the condi-tion supµ∈C

∫|x|>R |x|

p′

dµ(x) < +∞ for some p′> p implies relative compactness

of C in Pp(Rd). We will use these facts in Section 3.3.

80

Section 3.2

The Wasserstein-2 metric is sequentially lower semi-continuous with respect tothe weak convergence in Cb(Rd)∗ (see [5] Proposition 7.1.3). That is, for any twosequences of measures (µ1

n)n and (µ2n)n in P2(Rd) such that µjn → µj weakly∗ for

j = 1, 2 we haveW2(µ1, µ2) 6 lim inf

n→+∞W2(µ1

n, µ2n). (3.2.24)

Next we recall some facts about absolutely continuous curves in a metric space(X, d). A curve (a, b) 3 t 7→ γt ∈ X is said to be absolutely continuous oforder p ∈ [1,+∞), i.e. of class ACp((a, b);X), if there is a non-negative functionm ∈ Lp((a, b);L1) (the space of real valued p-integrable Borel functions withrespect to the one dimensional Lebesgue measure defined on (a, b)) such that

d(γt1 , γt2) 6∫ t2

t1

m(r) dr for t1, t2 ∈ (a, b) (3.2.25)

We then also denote γ ∈ ACp((a, b);X), and if this holds then the right metricderivative

limh→0

d(γt+h, γt)h

=: |γ|(t) (3.2.26)

exists for L1 a.e. t ∈ (a, b), with |γ| ∈ L2((a, b);L1). In addition |γ|(t) is thesmallest non-negative function m such that (3.2.25) holds with m = |γ|. For theproof of these facts we refer to [5] Theorem 1.1.2. If t 7→ γt is defined on (a, b)where −∞ 6 a < b 6 +∞ and (3.2.25) holds on (a′, b′) for all a < a′ < b′ < bthen we say that (a, b) 3 t 7→ γt is locally absolutely continuous of order p, i.e. ofclass ACploc((a, b);X).

The Wasserstein-2 space has the nice geometric property of being positivelycurved in the sense of Alexandrov (see [5] Theorem 7.3.2). It is well know that suchmetric spaces satisfy the first variation formula, which implies that various calculusformulas which hold on Riemannian manifolds can be recovered, in Wasserstein-2spaces. Also, the Euclidean cones over spaces of directions reflect the geometricstructure of the space (for the basic theory of geometry on spaces with curvaturebounded below or above in the sense of Alexandrov we refer to [20]). Howeversuch differential calculus on Alexandrov spaces is defined in the abstract way, andthere is a more concrete description of the tangent spaces in P2(Rd). This wasformally observed in the seminal paper [85], and a rigorous mathematical theoryis given in [5] through the ’continuity equation’ and its corollaries.

If µ : (a, b)→ P2(Rd) is of class AC2((a, b);P2(Rd)) then there is a Borel vectorfield (t, x) 7→ vt(x) defined L1 a.e. on (a, b) such that

vt ∈ L2(µt; Rd) L1 -a.e. on Rd (3.2.27)

with moreover ∫ b

a

||vt||2L2(µt;Rd) dt < +∞ (3.2.28)

and such that the following continuity equation holds∫ b

a

∫Rd

(∂tϕ(t, x) + 〈∇xϕ(t, x), vt(x)〉) dµt(x) dt = 0 (3.2.29)

81



for all ϕ ∈ C∞c ((a, b) × Rd). This claim and its converse are proved in [5] Theo-rem 8.3.1, and (3.2.29) is called the continuity equation. Shortly (3.2.29) can beexpressed by

∂tµt +∇ · (vtµt) = 0 in D′((a, b)× Rd). (3.2.30)

If the curve t 7→ µt is defined by means of transport i.e. µt := Stµ where (St)t>0 isa contraction semigroup on Rd given by (3.2.5), where b : Rd → Rd is a maximalmonotone operator, then the identity (3.2.29) is directly recovered with vt(x) :=−b(Stx) for all t > 0, x ∈ Rd and for each initial measure µ. Notice that (3.1.1)can be written in the following way:

∂tρt = ∇ · ((∇ρtρt

+ b)ρt) (3.2.31)

provided that µt = ρt · Ld is absolutely continuous for L1-a.e. t > 0.The (regular) tangent space of a measure µ ∈ P2(Rd) with (P2(Rd),W2) as the

ambient space is defined by

Tanµ(P2(Rd)) := ∇ϕ|ϕ ∈ C∞c (Rd)L2(µ;Rd)

(3.2.32)

and when µ Hd−1 (the (d − 1) dimensional Hausdorff measure on Rd) thisdefinition coincides with the abstract definition of the Euclidean tangent cone toa metric space with non-positive curvature in the sense of Alexandrov (see [5]Section 12).

Moreover, there is an L1 a.e. unique choice of the vector field (vt)t∈(a,b) in(3.2.29) such that vt ∈ Tanµt(P2(Rd)) for L1 a.e. t ∈ [a, b]. A path of such a curvecan also be locally ’linearized’, a claim comparable with the first order Taylorexpansion (see [5] Proposition 8.4.6 and Theorem 8.4.7).

As we already explained in Chapter 1, a lower semi-continuous mapping

ϕ : P2(Rd)→ (−∞,+∞], ϕ 6≡ +∞, (3.2.33)

which is bounded from below on some ball in P2(Rd) and is convex along gen-eralised geodesics (see [5] Chapter 9 for the definition) induces a gradient flowsemigroup on (P2(Rd),W2). Such a semigroup enjoys contraction and path regu-larity properties which are analogous to the properties of gradient flow semigroupson Hilbert spaces. Moreover, one defines the metric slope of such a functional ϕfor µ ∈ D(ϕ) := µ ∈ P2(Rd) : ϕ(µ) < +∞ by

|∂ϕ|(µ) := lim infσ→µ

(ϕ(µ)− ϕ(σ))+

W2(µ, σ)6 +∞ (3.2.34)

and this quantity corresponds to the quantity |−∇ϕ| in the Hilbert space setting.Its domain µ ∈ P2(Rd) : |∂ϕ|(µ) < +∞ corresponds to the domain of thegenerator of a contraction semigroup on a Hilbert space, and moreover

D(|∂ϕ|)W2 = D(ϕ)

W2 (3.2.35)

82

Section 3.2

holds. The semigroup (Rt)t>0 associated to ϕ is defined by the exponential for-mula, i.e. as the following limit

Rtµ := limn→+∞

(J tn

)nµ0, µ0 ∈ D(ϕ). (3.2.36)

The resolvents Jt appearing in (3.2.36) are defined as unique minimisers as follows:

Jtµ0 := argminν∈P2(Rd)

(12tW 2

2 (µ0, ν) + ϕ(ν)), t > 0, µ0 ∈ P2(Rd),

(3.2.37)see [5] Theorem 4.0.4. For each t > 0 the mapping Jt : D(|∂ϕ|) → D(|∂ϕ|) iscontinuous (see [5] Theorem 4.1.2). Moreover the Moreau-Yosida regularisation

ϕt(µ) :=12W 2

2 (µ,Jtµ) + ϕ(Jtµ) t > 0, µ ∈ P2(Rd), (3.2.38)

of such ϕ is jointly continuous on (0,+∞) × P2(Rd) (see [5] Lemma 3.1.2). Gra-dient flow semigroups on P2(Rd) exhibit the regularising effect, i.e. for each

µ ∈ D(ϕ)W2 the corresponding path of the semigroup (Rt)t>0 associated to ϕ

is of class AC2loc((0,+∞); (P2(Rd),W2)) and it is metrically differentiable from

the right at each t > 0. Also for any µ ∈ D(ϕ)W2 , |∂ϕ|(Rtµ) < +∞ holds for eacht > 0 and we have the following estimate (see [5] (4.3.4)):

|∂ϕ|(Rtµ) 61tW2(µ,Rtµ). (3.2.39)

The counterpart of the abstract Cauchy problem associated to a non-lineargenerator defined on a subset of a Hilbert space, can be formulated in this settingas the so called evolution variational inequality (EVI). That is, one searches acontinuous curve [0,+∞) 3 t 7→ µt ∈ P2(Rd) of class AC2

loc((0,+∞);P2(Rd))such that

ddt

12W 2

2 (µt, σ) + ϕ(µt) 6 ϕ(σ) for L1-a.e. t > 0, ∀σ ∈ D(ϕ). (3.2.40)

If a convex functional ϕ is defined on a Hilbert space, then the curves that satisfythe evolution variational inequalities as in (3.2.40), are precisely the paths of thesemigroup which is induced by the abstract Cauchy problem associated to ϕ. Inspite of being somewhat abstract and technical, the theory of gradient flows onWasserstein spaces turned out to be a very useful and novel way to interpretvarious classes of PDE’s, in particular the heat equation, the symmetric Fokker-Planck equation and the porous medium equation.

Let us now recall the Relative Entropy functional (with respect to Ld), whichis defined by

H(µ) :=

∫Rd

dµdLd log dµ

dLd dx if µ Ld

+∞ otherwise .(3.2.41)

83



We haveD(H)

W2 = P2(Rd) (3.2.42)

and a measure µ ∈ P2(Rd) satisfies µ ∈ D(|∂H|) if and only if

µ Ld, ρ :=dµdLd

∈W1,1(Rd) and ∇ρ = wρ (3.2.43)

for some vector field w ∈ L2(µ; Rd). In this case

|w|L2(µ;Rd) = |∂H|(µ) (3.2.44)

holds as well (see [5] Theorem 10.4.6.).Let us also mention that since H is convex along generalized geodesics (see [5]

Proposition 9.3.9) the metric slope of H is lower semi-continuous (see [5] Corollary2.4.10), i.e.

|∂H|(µ) 6 lim infn|∂H|(µn) if µn → µ. (3.2.45)

For the full treatment in much greater generality of these and related topics werefer to [5], [54], [85], [107], [106], and the numerous references therein.

3.3 Construction of the semigroup on (P2,W2) –The Trotter-Kato product formula

The primary aim of this section is to show that (3.1.3) holds. It is worth noticingthat proving this product formula requires a different approach than in the classi-cal case of the linear C0-semigroups or the product formula for maximal monotoneoperators on Hilbert spaces and not merely due to the fact that the underlyingspace P2(Rd) lacks the linear structure. In Chapter II we have proved such a for-mula for gradient flows on CAT(0) spaces which establishes that the non-positivecurvature condition assumed there in fact suffices. In [5], the authors discoveredthat geodesics in P2(Rd) are (−1)-convex along the so called generalized geodesics,a crucial property used in their proofs of the variational estimates and in the con-struction of the flow associated to convex functionals that do not posses localLipschitz properties.1

One might attempt to follow the approach of Chapter II, since the space(P2(Rd),W2) enjoys the (−1)-convexity along generalized geodesics of W 2

2 , andthis property suffices for the construction of the solutions of the Evolution Varia-tional inequality (see [5]). However, this property of W 2

2 does not seem to sufficefor proving the product formula for gradient flows on (P2(Rd),W2) (see Remark3.3.14).

1 We mention that flows associated to locally Lipschitz convex functionals defined on metricspaces with one sided curvature bound can be constructed as in [86] and [69]. Although suchsituations are interesting from the point of view of the theory of metric geometry, functionalswhose subdifferential corresponds to a partial differential operator are typically only lower semi-continuous.

84

Section 3.3

We will take an approach where we exploit the particular form of our equation(3.1.1) as follows. We will first show that the semigroups (P 1

t )t>0,(P 2t )t>0 and

(Qt)t>0 defined in (3.2.12), (3.2.13) and (3.2.9), respectively, are C0-semigroupson C0(Rd). We emphasize that in proving these facts we do not assume the drift bin equation (3.1.1) to be Lipschitz or even Holder (and in particular the standardHolder theory as in [67] does not apply), but only the monotonicity (3.2.1) and thelinear growth condition (3.2.3). Next we show that the product formula holds for(P 1t )t>0 and (P 2

t )t>0 in C0(Rd), the ’sum’ semigroup being (Qt)t>0. The Lipschitzassumption of b is used in this step only of the proof, for the reason that in this casethe linear subspace C∞c (Rd) is known to be a core for the generator of (Qt)t>0, sothat we can apply the (linear) Trotter product formula for the backward equation(3.2.11). If one can show that C∞c (Rd) is a core by other means (for instance it isnot hard to show that the variation-of-constants formula holds in this case), thenthe Lipschitz condition (3.2.2) may be replaced by the much weaker linear growthcondition (3.2.3) in the Trotter product formula.

The product formula for the backward equation (3.2.11) implies that conver-gence of iterations as in (3.1.3) holds in the weak∗ sense in C0(Rd)∗, since the semi-groups (R1

t )t>0 and (R2t )t>0 are the dual semigroups of the semigroups (P 1

t )t>0

and (P 2t )t>0, respectively. Next we will obtain bounds for the fourth order mo-

ments of iterations (R2t/nR

1t/n)kµ for µ ∈ P4(Rd), and show (3.1.3) with the aid of

this result.Let us start executing our program by showing that the semigroups (P 1

t )t>0,(P 2t )t>0, and (Qt)t>0, defined in (3.2.12), (3.2.13), and (3.2.9) respectively (where

b is a maximal monotome subset of Rd × Rd), are C0-semigroups on C0(Rd).The next result is standard and not difficult to prove.

Proposition 3.3.1. The restriction of (P 2t )t>0 to C0(Rd) is a contraction C0-

semigroup.

Recall Lemma 3.2.1 from the previous section.

Proposition 3.3.2. Assume that b : Rd → Rd is continuous and that it satisfiesconditions (3.2.1) and (3.2.3). Then the restriction of (P 1

t )t>0 to C0(Rd) is acontractive C0-semigroup.

Proof. Let us first show that C0(Rd) is invariant under the action of P 1t for each

t > 0. Fix t > 0 and f ∈ C0(Rd). Since St is a contraction on Rd, we clearly havethat P 1

t f is a continuous function.Next we show that if |x| → +∞ then P 1

t f(x)→ 0. This will hold if |Stx| → +∞whenever |x| → +∞. Fix any x ∈ Rd, and denoting xt := Stx observe that by(3.2.5) and (3.2.3)

|xt| = | − b(xt)| 6 C(1 + |xt|) (3.3.1)

holds L1-a.e. on [0,+∞) where xt denotes the L1-a.e. defined derivative of thecurve t→ xt. Thus for L1-a.e. t > 0 we have that

− ddt

log(1 + |xt|) =− d

dt |xt|1 + |xt|

6|xt|

1 + |xt|6 C (3.3.2)

85



where the first inequality in (3.3.2) is seen to hold by means of the differencequotients. Integrating (3.3.2) between 0 and t we obtain |log 1+|xt|

1+|x| | > −Ct, hence

|xt| > (1 + |x|)e−Ct − 1, (3.3.3)

which implies that for any sequence (xn)n ⊂ Rd, |xn| → +∞ imples that |P 1t xn| →

+∞, for each t > 0. Therefore P 1t f ∈ C0(Rd) for each t > 0. The semigroup

property holds for (P 1t )t>0 by the semigroup property of (St)t>0, while the con-

traction property holds since |f St| obviously can not assume values larger than|f | itself. Finally in order to show the strong continuity property of (P 1

t )t>0,which is that limt↓0 P

1t f = f holds in C0(Rd), we apply [94] Lemma III 6.7. This

lemma asserts that a contractive semigroup on C0(Rd) is strongly continuous iflimt↓0 P

1t f(x) = f(x) holds for each x ∈ Rd. But this claim obviously holds in

our case since for each x ∈ Rd we have limt↓0 Stx = x for each x ∈ Rd and f is acontinuous function.

Next we show that (Qt)t>0 is a contraction C0-semigroup on C0(Rd) with theaid of (3.2.7). Recall that (Xx

t )t>0 denotes the up to indistinguishability uniquesolution of (3.2.7) with deterministic initial condition Xx

0 = x for x ∈ Rd.

Proposition 3.3.3. Assume that b : Rd → Rd is continuous and that it satisfiesconditions ((3.2.1) and 3.2.3). Then the restriction of (Qt)t>0 to C0(Rd) is acontraction C0-semigroup.

Proof. Fix f ∈ C0(Rd).Let us first show that Qtf is a continuous function on Rd for t > 0. For

fixed x, y ∈ Rd observe that paths of the difference process Xxt − X

yt are locally

absolutely continuous and we have the following estimate

ddt|Xx

t −Xyt |2 = −〈b(Xx

t )− b(Xyt ), Xx

t −Xyt 〉 6 0, ∀t > 0 (3.3.4)

due to assumption (3.2.1). Hence for each t > 0 and for all x, y ∈ Rd we have that

|Xxt −X

yt | 6 |x− y|, a.s. (3.3.5)

Since each function in C0(Rd) is uniformly continuous (3.3.5) implies that Qtf isalso (uniformly) continuous for each t > 0.

Next we show that Qtf vanishes at infinity. Since for any x ∈ Rd the processXxt is continuous a.s., the process Xx

t −√

2Wt = −∫ t

0b(Xx

s ) ds has everywhere in[0,+∞) differentiable paths a.s. and using (3.2.3) we obtain

| ddt

(Xxt −√

2Wt)| = | − b(Xxt )| 6 C(1 + |Xx

t |) ∀t ∈ [0,+∞) a.s. (3.3.6)

Moreover we have (1 + |Xxt |) − (1 + |Xx

t −√

2Wt|) 6√

2|Wt| for t > 0 a.s., thusalso

1 + |Xxt |

1 + |Xxt −√

2Wt|6

√2|Wt|

1 + |Xxt −√

2Wt|+ 1 6

√2|Wt|+ 1 (3.3.7)

86

Section 3.3

for t > 0, a.s. Now with the aid of (3.3.6) and (3.3.7) we estimate

| ddt

log(1 + |Xxt −√

2Wt|)| =| ddt |X

xt −√

2Wt||1 + |Xx

t −√

2Wt|6| ddt (X

xt −√

2Wt)|1 + |Xx

t −√

2Wt|

6C(1 + |Xx

t |)1 + |Xx

t −√

2Wt|6 C(

√2|Wt|+ 1),

(3.3.8)

and integrating (3.3.8) between 0 and t gives that log 1+|Xxt −√

2Wt|1+|x| > −Ct −

C√

2∫ t

0|Ws|ds for t > 0 a.s. for each x ∈ Rd. Thus we obtained that for any

x ∈ Rd|Xx

t −√

2Wt| > (1 + |x|)e−Ct−C√

2R t0 |Ws| ds − 1 (3.3.9)

holds for each t > 0 a.s.In order to show that Qtf vanishes at infinity fix a sequence (xn)n ⊂ Rd such

that |xn| → +∞. We must show that Qtf(xn) → 0. Due to (3.3.9) there is ameasurable subset Ω of the probability space on which (Wt)t>0 and (Xxn

t )t>0 aredefined, of measure 1 such that (3.3.9) holds on Ω for each xn. Since for eachω ∈ Ω and for each t > 0 the quantity e−Ct−C

√2R t0 |Ws| ds is constant and finite

(3.3.9) implies that |Xxnt | → +∞ on Ω. Therefore since f ∈ C0(Rd), definition

(3.2.9) implies that Qtf ∈ C0(Rd) holds for each t > 0.The contraction property is clear from (3.2.9) and the semigroup property

follows by the standard arguments (i.e. the integrator (Wt)t>0 is a Levy prosess,and we have the weak uniqueness for (3.2.7)). For the strong continuity of the pathswith respect to the C0(Rd) norm, we again recall that by [94] Lemma III 6.7 weonly need to check pointwise continuity, which is immediately clear from definitionof (Qt)t>0 and a.s. continuity of solutions (Xx

t )t>0. The proof is completed.

An easy computation gives

ddt|t=0f(Stx) = −〈∇f(x), b(x)〉

if f is a C1 function. Hence the generator A1 of (P 1t )t>0 satisfies

A1f = −〈∇f, b〉 (3.3.10)

for any f ∈ C0(Rd)⋂C1(Rd) such that the function at the right side in (3.3.10)

is a member of C0(Rd). Likewise the generator A2 of (P 2t ) satisfies

A2f = ∆f (3.3.11)

for any f ∈ C2(R)⋂C0(Rd) such that ∆f ∈ C0(Rd). Now we can prove the

product formula for the equation (3.2.8).

Proposition 3.3.4. Assume that b : Rd → Rd satisfies (3.2.1) and (3.2.2). Thenfor any f ∈ C0(Rd)

limn→+∞

(P 2t/nP

1t/n)nf = Qtf in C0(Rd), (3.3.12)

87



andlim

n→+∞(P 1t/nP

2t/n)nf = Qtf in C0(Rd), (3.3.13)

both uniformly on compact time intervals.

Proof. Let A denote the generator of (Qt)t>0. Notice that C∞c (Rd) is a subset ofD(Aj) for j = 1, 2 which is moreover dense in C0(Rd). In light of [39] Chapter 1Corollary 6.7 both formulas (3.3.12) and (3.3.13) hold if there is a subsetD ⊂ D(A)which is a core for A such that D ⊂ D(A1)

⋂D(A2) and

Af = A1f +A2f ∀f ∈ D (3.3.14)

Since we assumed b to be Lipschitz continuous [39] Chapter 8 Theorem 2.5 guar-antees that the closure in the graph norm of (f,Af) with f ∈ C∞c (Rd) is singlevalued and generates a C0-semigroup on C0(Rd). Since this closure is containedin the graph of A, and due to Proposition 3.3.3 is also a generator of the C0-continuous semigroup (Qt)t>0, these two objects must coincide, and C∞c (Rd) isindeed a core for A, and we have proved both claims.

Corollary 3.3.5. Under the same conditions as in Proposition 3.3.4, for µ ∈M(Rd) = C0(Rd)∗

(R2t/nR

1t/n)nµ→ Rtµ for each t > 0, (3.3.15)

(R1t/nR

2t/n)nµ→ Rtµ for each t > 0, (3.3.16)

the convergence being in the weak* sense of C0(Rd)∗.

Proof. Follows by definition.

Remark 3.3.6. Observe that we have indeed used the Lipschitz assumption(3.2.2) of b instead of linear growth assumption (3.2.3) only in order to showthat C∞c (Rd) is a core for A. This stronger assumption will not be used in theremainder of the section either, thus finding an alternative way to show that someset of smooth functions is a core for (Qt)t>0, or alternatively showing that thissemigroup preserves smoothness will yield all of the product formulas in this sec-tion under the assumptions in Proposition 3.3.3.

Let us now proceed with the analysis of the dual semigroups (R1t )t>0 and

(R2t )t>0. Notice that if b : Rd → Rd satisfies (3.2.3) then for any µ ∈ P2(Rd) we

have that b ∈ L2(µ; Rd).

Lemma 3.3.7. Assume that b : Rd → Rd is continuous and that it satisfies(3.2.3) and (3.2.1). Then P2(Rd) is invariant under the action of the semigroup(R1

t )t>0 and this restriction is a W2-contraction. Moreover for any µ0 ∈ P2(Rd)and 0 < t1 < t2

W 22 (µt1 , µt2) 6 (t2 − t1)2

∫|b(x)|2 dµ0(x) (3.3.17)

where µt := R1tµ0.

88

Section 3.3

Proof. Pick any µ0 ∈ P2(Rd) and any t > 0 and compute∫|x|2 dµt =

∫|Stx|2 dµ0(x) 6 2|St0|2 + 2

∫|Stx− St0|2 dµ0(x)

6 2|St0|2 + 2∫|x|2 dµ0(x) < +∞

so P2(Rd) is (R1t )t>0 invariant. Next for µ1, µ2 ∈ P2(Rd) pick an optimal transport

plan γ ∈ Γo(µ1, µ2). As (St, St)#γ ∈ Γ(Stµ1, Stµ2) holds for each t > 0 and (St)t>0

is a contractive semigroup on Rd we can estimate

W 22 (R1

tµ1, R1tµ2) 6

∫|Stx− Sty|2 dγ(x, y) 6

6∫|x− y|2 dγ(x, y) = W 2

2 (µ1, µ2)(3.3.18)

hence (R1t )t>0 is a W2-contraction.

Finally for 0 < t1 < t2 and µ0 ∈ P2(Rd) the map ψ(x) := (St1x, St2x) satisfiesψ#µ0 ∈ Γ((St1)#µ0, (St2)#µ0) hence we estimate

W 22 (R1

t1µ0, R1t2µ0) 6

∫|St1x− St2x|2 dµ0(x) =

=∫

Rd

∣∣∣∣∫ t2

t1

−b(Ssx) ds∣∣∣∣2 dµ0(x) 6 |t0 − t1|2

∫|b(x)|2 dµ0(x)

(3.3.19)

by (3.2.5) and (3.2.6).

Recall the relative entropy functional defined in (3.2.41) and let us state thefollowing result from [54] and [5].

Theorem 3.3.8. P2(Rd) is invariant under the action of (R2t )t>0 and this re-

striction is a W2-contraction. Moreover for any initial measure µ0 ∈ P2(Rd) thepath t→ R2

tµ0 is of class AC2loc((0,+∞), (P2(Rd),W2)) and we also have that

H(R2tµ0) < +∞ , |∂H|(R2

tµ0) < +∞ ∀ t > 0. (3.3.20)

Furthermore t → H(R2tµ0) and t → |∂H|(R2

tµ0) are non-increasing functions. Ifµ0 ∈ D(|∂H|) then the path 0 6 t 7→ R2

tµ0 is Lipschitz on [0,+∞). Its metricderivative at time t equals |∂H|(R2

tµ0), and in particular it is non-increasing.

In the next two lemmas we obtain estimates for the fourth order moments ofthe successive iterations of a measure µ0 ∈ P4(Rd).

Lemma 3.3.9. Assume that b : Rd → Rd is continuous and that it satisfiesconditions (3.2.1) and (3.2.3). Then there is a constant c > 0 depending only onthe linear growth constant of b, such that for any µ0 ∈ P4(Rd) and for any t > 0∫

|x|4 dR1tµ0(x) 6 ect

∫|x|4 dµ0(x) + ctect. (3.3.21)

89



Proof. By (3.2.3), (3.2.5) and (3.2.6) we estimate∫Rd|x|4 dR1

tµ0(x) =∫

Rd|Stx|4 dµ0(x) =

∫Rd

∣∣∣∣x− ∫ t

0

b(Ssx) ds∣∣∣∣4 dµ0

=∫

Rd(|x|2 − 2

⟨x,

∫ t

0

b(Ssx) ds⟩

+∣∣∣∣∫ t

0

b(Ssx) ds∣∣∣∣2)2 dµ0(x)

6∫

Rd(|x|4 + 4|x|3tC(1 + |x|) + 4|x|(tC)3(1 + |x|)3

+ 2|x|2(tC)2(1 + |x|)2 + 2|x|2(tC)2(1 + |x|)2 + (tC)4(1 + |x|)4) dµ0(x)

6∫

Rd(|x|4 + 4tC|x|3 + (4tC)2|x|2 + (4tC)3|x|+ (4tC)4+

+ (4tC + (4tC)2 + (4tC)3 + (4tC)4)|x|4) dµ0(x).(3.3.22)

Since for k = 1, 2, 3 we can write∫Rd|x|k dµ0(x) =

(∫|x|>1

+∫|x|<1

)|x|k dµ0(x) 6 1 +

∫|x|4 dµ0(x),

(3.3.22) implies∫Rd|x|4 dR1

tµ0(x) 6∫

Rd

4∑k=0

(4tC)k|x|4 dµ0(x) +4∑k=0

(4tc)k

6 ect∫

Rd|x|4 dµ0(x) + ctect

for a finite constant c > 0.

In a similar fashion we have the following lemma.

Lemma 3.3.10. There is a constant c > 0 such that for each µ0 ∈ P4(Rd) andfor each t > 0 we have that∫

|x|4 dR2tµ0(x) 6 ect

∫|x|4 dµ0(x) + ct (3.3.23)

Proof. This is a well known fact. It follows for instance by [104] where a similarestimate is proven for the Wasserstein resolvent associated to the relative entropyfunctional of any initial measure with finite fourth order moment—then the expo-nential formula (4.0.11) in [5] implies our claim.

Lemma 3.3.11. Assume that b : Rd → Rd is continuous and that it satisfiesconditions (3.2.1) and (3.2.3). Then for any T > 0 there are constants c, C > 0such that for each µ0 ∈ P4(Rd), 0 6 t 6 T , n ∈ N and 0 6 k 6 n the followingestimates hold∫

Rd|x|4 d(R2

t/nR1t/n)kµ0(x) 6 e2cT

∫Rd|x|4 dµ0(x) + C, (3.3.24)

90

Section 3.3

∫Rd|x|4 d(R1

t/nR2t/n)kµ0(x) 6 e2cT

∫Rd|x|4 dµ0(x) + C. (3.3.25)

Proof. Let c > 0 be a constant such that the conclusions of Lemma 3.3.9 andLemma 3.3.10 hold. By induction we have that for k = 1, 2, ..., n∫

Rd|x|4 d(R2

t/nR1t/n)kµ0(x) 6 ek·2ct/n

∫Rd|x|4 dµ0(x) + ct

2k∑j=1

ejct/n

6 e2cT

∫Rd|x|4 dµ0(x) + cT

2n∑k=1

ekcT/n

= e2cT

∫Rd|x|4 dµ0(x) +

cT

necT/n

1− e2cT

1− ecT/n

and as the function x 7→ xex

ex−1 in continuous on [0, T ] and 0 < Tn 6 T for each

n ∈ N there exists a constant C > 0 such that (3.3.24) holds. The claim in (3.3.25)holds by the symmetry.

With the aid of Corollary 3.3.5 and Lemma 3.3.11 we are now going to provethe product formula (3.1.3). Notice that in particular it holds when b is a gradientof a convex l.s.c. function V on Rd, provided that it moreover satisfies (3.2.2).

Theorem 3.3.12. Assume that b : Rd → Rd satisfies conditions (3.2.1) and(3.2.2). Then for each t > 0 and µ0 ∈ P2(Rd) the following product formulas hold:Rtµ0 ∈ P2(Rd) and

(R2t/nR

1t/n)nµ0

W2−→ Rtµ0, (R1t/nR

2t/n)nµ0

W2−→ Rtµ0. (3.3.26)

Consequently the semigroup (Rt)t>0 is a W2-contraction.

Proof. We will only show the first claim in (3.3.26) since the second claim followsby symmetry. Fix T > 0 and pick any µ0 ∈ P4(Rd). By Corollary 3.3.5 we havethat (R2

t/nR1t/n)nµ0 converges to Rtµ0 in the weak* sense in C0(Rd)∗. Moreover by

Lemma 3.3.11 the fourth order moments of these iterations are uniformly boundedfor n ∈ N and t ∈ [0, T ], which implies that the second moments are uniformlyintegrable (see Section 3.2 below (3.2.23)). This means that for each t ∈ [0, T ]any subsequence of (R2

t/nR1t/n)nn has a further subsequence which converges

with respect to the W2 distance to some measure in P2(Rd). But since W2 con-vergence implies weak∗ convergence in duality with Cb(Rd), it also implies weak∗

convergence in C0(Rd)∗ because the latter space contains less functions for whichthe convergence of the integrals must be tested. But by the convergence of theiterations in C0(Rd)∗ there can be only one limit and that limit is Rtµ0. SinceT > 0 was arbitrary this implies that

(R2t/nR

1t/n)nµ0

W2−→ Rtµ0 as n→ +∞, ∀µ0 ∈ P4(Rd) , ∀t > 0. (3.3.27)

91



Moreover, Lemma 3.3.7 and Theorem 3.3.8 imply that for each n the mapping(R2

t/nR1t/n)n : P2(Rd) → P2(Rd) is a W2-contraction thus (3.3.27) implies that

Rt : P4(Rd)→ P2(Rd) is a W2-contraction, too.For a general µ0 ∈ P2(Rd) pick ε > 0 and µ1 ∈ P4(Rd) such thatW2(µ0, µ1) < ε

(we can always find such µ1). Since (R2t/nR

1t/n)nµ1

W2−→ Rtµ1, this sequence isCauchy hence there is an N ∈ N such that for n,m > N

W2((R2t/nR

1t/n)nµ1, (R2

t/mR1t/m)mµ1) < ε.

But then since (R2t/nR

1t/n)n is a W2- contraction for each n ∈ N and for each t > 0

a twofold application of the triangle inequality yields

W2((R2t/nR

1t/n)nµ0, (R2

t/mR1t/m)mµ0) < 3ε, n,m > N,

i.e. this sequence is Cauchy. The limit must be the same as the weak∗ limit Rtµ0 ofthese iterations, and we have proven that the product formula (3.3.26) holds. TheW2-contraction property follows directly since (R2

t/nR1t/n)n is a W2-contraction for

each t > 0 and for each n ∈ N.

Remark 3.3.13. Arguing as in Lemma 3.3.9, Lemma 3.3.10 and Lemma 3.3.11we can obtain that for each T > 0 and µ ∈ P2(Rd)

sup∫

Rd|x|2 d(R2

t/nR1t/n)kµ(x)|n ∈ N, k = 1, 2, ..., n, t ∈ [0, T ] < +∞ (3.3.28)

i.e. such set of iterations is contained in a ball in the metric space P2(Rd) ifb : Rd → Rd is continuous and satisfies conditions (3.2.1) and (3.2.3). We will usethis fact in the following section.

Remark 3.3.14. As we mentioned at the begining of this section, in light ofthe results of Chapter II, it might seem plausible that the (−1)-convexity alongthe generalized geodesics of W 2

2 together with the convexity along generalizedgeodesics and lower semi-continuity of the functionals under consideration, mightbe sufficient to prove the product formula. However, attempts by the author ofsuch a ’direct’ translation of the results of Chapter II did not succeed. The majorobstacle seems to be the following: Trotter-Kato approximations approximate indirections of flows or resolvents of two (or more generally n) functionals, thus ifwe adopt the same philosophy as in [5] then in each of these steps the positivecurvature of P2(Rd) should be compensated somehow. And combining the (atleast) two steps together seems to require a stronger convexity assumption thanthe convexity along generalized geodesics. More precisely it seems that assumingconvexity along interpolation curves induced by a certain kind of 5-plans, could besufficient to construct the approximate resolvents and semigroups (see Chapter II).However, there may not be relevant functionals that posses such strong convexityproperties. In fact, there are not many known geodesically convex functionals onP2(Rd).

92

Section 3.4

3.4 Absolute continuity of paths and the regular-ising effect

As it is shown in [5] gradient flow paths of a semigroup (Tt)t>0 associated toa functional ϕ on P2(Rd) that has appropriate convexity, non-degeneracy andlower semi-continuity properties exhibits very strong path regularity properties.Precisely, for any initial point µ0 ∈ D(ϕ) ⊂ P2(Rd) the curve t 7→ Ttµ0 = µt isLipschitz on [ε,+∞) for each ε > 0, and µt ∈ D(|∂ϕ|) for all t > 0. In such asituation the analogy with semigroups induced by a l.s.c. convex functional on aHilbert space, or with linear C0-semigroups on a Banach space, is that the domainof the metric slope (3.2.34) corresponds to the domain of the subdifferential orthe domain of the linear generator, respectively, and locally Lipschitz paths on[0,+∞) with values in P2(Rd) correspond to locally Lipschitz paths on [0,+∞)with values in a Hilbert space, or in a reflexive Banach space, respectively.

However since b is not assumed to be a gradient there is no reason why oursemigroup (Rt)t>0 should be a gradient flow semigroup on P2(Rd), and we cannotapply theory from [5] to conclude that (Rt)t>0 exhibits the regularising effect.Yet if b satisfies (3.2.1) and (3.2.3), the paths of the semigroup (R1)t are Lipschitzas we showed in Lemma 3.6. We can interpret this observation as its ’generator’having full domain. We have proved the product formula in the previous sectionand loosely speaking this means that (Rt)t>0 is induced by the ’sum of generators’of semigroups (R1

t )t>0 and (R2t )t>0. Therefore it seems reasonable to expect that

perturbing (in the sense of ’addition’) a ’generator’ with another ’generator’ definedon the whole space should not affect any regularity properties. That is, we expectthat (Rt)t>0 has the same locally Lipschitz path properties as (R2

t )t>0 does, andthat for any initial measure µ0 we should have that µt = Rtµ0 ∈ D(∂H) for eacht > 0, where H is the relative entropy functional defined in (3.2.41).

The proof of the informal analysis will be carried out in two steps and splitinto two separate propositions. We are going to apply our product formula provenin Theorem 3.3.12, and use the variational techniques and results presented in[5]. However we do not have a functional that generates the semigroup (Rt)t>0.A major difficulty is to establish that the locally Lipschitz property is preservedby the limit of the curves obtained by the product approximations (see (3.4.5)below). The fact that the operators (R2

tnR1

tn

)n are W2-contractions turnes out tobe crucial, and a product formula where the resolvents associated to H are usedinstead of the semigroup (R2

t )t>0 (such as the authors considered in [27]) does notsuffice.

Throughout the remainder of this section we will use the following notation. Ifb : Rd → Rd satisfies the Lipschitz condition (3.2.2) then we denote its Lipschitzconstant by l0 := Lip b := supx6=y

|b(x)−b(y)||x−y| and we also set l := dl0. Recall

moreover that (3.2.2) implies (3.2.3). For simplicity of the notation we also denoteR2,1s := R2

sR1s for s > 0. Recall also the relative entropy functional H defined in

(3.2.41). Before proceeding we state a result from [79].

Theorem 3.4.1. Assume that b satisfies (3.2.1) and (3.2.2). Then for any t > 0

93



and µ0 ∈ D(H) we haveH(R1

tµ0) 6 H(µ0) + tl, (3.4.1)

thus alsoH(R2,1

t µ0) 6 H(µ0) + tl, (3.4.2)

andH(Rtµ0) 6 H(µ0) + tl. (3.4.3)

Proof. For (3.4.1) see [79] Lemma 5.7. (3.4.2) follows since (R2t )t>0 is a gradient

flow semigroup and therefore decreases the value of the entropy, see Theorem 3.7.(3.4.3) follows by Theorem 3.3.12, (3.4.2) and the lower semi-continuity of H:

H(Rtµ0) 6 lim infn→+∞

H((R2,1tn

)nµ0) 6 lim infn→+∞

(H(µ0) + n

t

nl

)= H(µ0) + tl. (3.4.4)

Proposition 3.4.2. Assume that b satisfies (3.2.1) and (3.2.2). Then for anyµ0 ∈ D(H) and for all ε > 0 the curve t 7→ µt := Rtµ0 is Lipschitz on [ε,+∞), inparticular it is of class AC2

loc((0,+∞);P2(Rd)). Moreover µt ∈ D(|∂H|) for eacht > 0.

Proof. Fix arbitrary µ0 ∈ D(H) and t > 0, and denote tn := t · 2−n for n ∈ N.Define piecewise constant curves on [0, t]

µn0 := µ0, µn(k+1)tn:= R2,1

tn µnktn , µnktn+h := µn(k+1)tn

,

for k = 0, ..., 2n − 1 and 0 < h < 2−n.(3.4.5)

Due to (2.4.26) in [5], t 7→ H(R2t ν) is decreasing for any ν ∈ P2(Rd) so that

integrating the EVI (3.2.40) gives

W 22 (R2,1

tn µ0, R1tnµ0)

2tn6 H(R1

tnµ0)−H(R2,1tn µ0) (3.4.6)

and together with (3.4.1) this yields

W 22 (R2,1

tn µ0, R1tnµ0)

2tn6 H(µ0)−H(R2,1

tn µ0) + tnl. (3.4.7)

Moreover arguing as in Lemma 3.3.7 (see (3.3.19)) we estimate

W 22 (R1

tnµ0, µ0) 6∫

R|x− Stnx|2 dµ0(x)

6 t2n

∫Rd|b(x)|2 dµ0(x) 6 C2t2n(1 +W 2

2 (µ0, δ0)),(3.4.8)

94

Section 3.4

where C depends on the linear growth constant of b and we employ identity(3.2.21). As

W 22 (R2,1

tn µ0, µ0) 6 2W 22 (R2,1

tn µ0, R1tnµ0) + 2W 2

2 (R1tnµn, µ0) (3.4.9)

combining (3.4.9), (3.4.7) and (3.4.8) yields

W 22 (R2,1

tn µ0, µ0)tn

6 4(H(µ0)−H(R2,1tn µ0))+4tnl+2C2tn(1+W 2

2 (µ0, δ0)). (3.4.10)

Recalling definition (3.4.5) we thus have for k = 0, 1, ..., 2n − 1

W 22 (µnktn , µ

n(k+1)tn

)

tn64(H(µnktn)−H(µn(k+1)tn

)) + 4tnl + 2C2tn

+2tnC2W 22 (µnktn , δ0)

(3.4.11)

and summing these inequalities for k = 0, 1, ..., 2n − 1 yields

2n−1∑k=0

W 22 (µnktn , µ

n(k+1)tn

)

tn64(H(µn0 )−H(µnt )) + 4lt+ 2C2t

+2n−1∑k=0

2C2tnW22 (µnktn , δ0).

(3.4.12)

As we observed in (3.3.28) the set of measures µns : n ∈ N, s ∈ [0, t] iscontained in a ball in P2(Rd). Moreover the proof of [5] Theorem 11.2.5 or alter-natively [26] Lemma 4.1 implies that infH(µnt )|n ∈ N > −∞. Hence there is aconstant K < +∞ such that the right hand side in (3.4.12) is bounded by K forall n ∈ N (recall that µn0 = µ0 for each n ∈ N).

Next, we introduce the piecewise constant (hence measurable) functions un on[0, t) for n ∈ N

un(0) := 0, un(s) :=W2(µnktn , µ

n(k+1)tn

)

tn(3.4.13)

for s = ktn +h, 0 < h 6 2−n, k = 0, ..., 2n− 1. These functions are non-increasingsince R2,1

s is a contraction for each s > 0.By definition (3.4.13) and the triangle inequality we have that for each n ∈ N,

and 0 6 s1 < s2 6 t such that s2 − s1 > 2−nt,

W2(µns1 , µns2) 6

∫ s2

s1

un(s) ds ∀n ∈ N, (3.4.14)

holds. Moreover ∫ t

0

(un(s))2 ds =2n−1∑k=0

W 22 (µnktn , µ

n(k+1)tn

)

tn(3.4.15)

95



so that (3.4.12) together with the discussion directly below (3.4.12) yields

supn

∫ t

0

(un(r))2 dr <∞. (3.4.16)

Now by Helly’s theorem (see [5] Theorem 3.3.3) there is a subsequence of unnwhich we again denote unn and a non-increasing function u : [0, t] → [−∞,∞]such that

un(r)→ u(r) ∀r ∈ [0, t]. (3.4.17)

By taking a further subsequence if necessary we may assume that un → u weaklyin L2((0, t);L1). Since u > 0 is non-decreasing and also in L2((0, t);L1), we havethat for each r ∈ (0, t], u(r) ∈ [0,+∞). Fix s1, s2 ∈ (0, t) of the form s1 = k12−n0t,s2 = k22−n0t for some k1, k2, n0 ∈ N with k1, k2 ≤ 2n0−1. Theorem 3.3.12 implies

µnsj = (R2,1t/2n)kj2

n−n0 tµ0 = (R tkj/2n0

kj2n−n0

)kj2n−n0

µ0 =

= (R2,1sj

kj2n−n0

)kj2n−n0

µ0W2−→ Rsjµ0 for j = 1, 2 ,

(3.4.18)

and since un → u weakly in L2((0, t);L1) we conclude

W2(µs1 , µs2) = limn→+∞

W2(µns1 , µns2)

6 limn→+∞

∫ s2

s1

un(r) dr =∫ t

0

u(r) dr.(3.4.19)

For general 0 6 s1 < s2 < t take sequences srj ↓ sj for j = 1, 2. By the weakcontinuity of solutions of (3.2.7) which we observed in Section 3.2 and (3.2.24) weconclude that (3.4.19) holds for all 0 < s1, s2 6 t. We have now established that0 < t 7→ Rtµ ∈ AC2

loc((0,+∞);P2(Rd)).Let us prove the locally Lipschitz property. Well since r 7→ u(r) is non-

increasing, for any s > t > 0 and h > 0 sufficiently small we have

W2(µs+h, µs) 6∫ s+h

s

u(r) dr 6∫ t+h

t

u(r) dr 6 hu(t) (3.4.20)

and

W2(µs, µs−h) 6∫ s

s−hu(r) dr 6

∫ s+t2

s+t2 −h

u(r) dr 6 hu(s+ t

2− h) 6 hu(t). (3.4.21)

Therefore lim suph→0W2(µs,µs+h)

h 6 u(t) for s > t > 0. Hence ε 6 t 7→ Rtµ0 isLipschitz for each ε > 0 and for each µ0 ∈ P2(Rd).

Finally to prove that |∂H|(Rtµ) < +∞ for t > 0 recall (3.2.39), which gives

|∂H|(R2,1t µ0) 6

1tW2(R1

tµ0, R2,1t µ0) 6

1tW2(R1

tµ0, µ0) +1tW2(µ0, R

2,1t µ0).

(3.4.22)

96

Section 3.4

In particular for each n0 6 n ∈ N, t > 0, k ∈ 0, 1, ..., 2n0 − 1 and s = (k +1)2−n0t 6 t, the curves [0, t] 3 r 7→ µnr defined in (3.4.5) satisfy

|∂H|(µns ) 61tnW2(R1

tnµnk2n0−ntn

, µnk2n0−ntn) +

1tnW2(µnk2n0−ntn

, µns )

6∫

Rd|b(x)|2 dµns (x) + un(s)

6 C2 + C2W 22 (µns , δ0) + un(s),

(3.4.23)

with the aid of (3.3.17). By (3.4.17), un(r) → u(r) < +∞ (for each r ∈ [0, t] andthis is where really we need Helly’s theorem) and µns → Rsµ0, hence by the lowersemi-continuity of |∂H| (see (3.2.45)) we deduce that

|∂H|(µs) 6 lim infn→+∞

|∂H|(µns ) 6 supn|∂H|(µns )

6 C2 + C2 supnW 2

2 (µns , δ0) + u(s) < +∞.(3.4.24)

In particular, choosing any n0 ∈ N and k := 2n0 − 1 (3.4.24) gives that

|∂H|(µt) < +∞. (3.4.25)

Since the curve r 7→ µr does not depend on the choice of t in (3.4.5), while t > 0was arbitrarily chosen there we conclude that (3.4.25) holds for each t > 0. Theproof is now completed.

We need another result from the PhD thesis [79] of L.Natile, where he provesthat for any initial µ ∈ D(|∂H|) (recall (3.2.43) for the characterisation) there is aunique distributional solution 0 6 t 7→ µt of (3.1.1) which is absolutely continuouson [0,+∞) and such that µt Ld for each t > 0. Natile calls such solutionsthe Wasserstein solutions (see [79] Definition 5.1 and Remark 5.2). Moreover inTheorem 5.4 of [79] Natile shows that

12W 2

2 (µt2 , σ)− 12W 2

2 (µt1 , σ) 6∫ t2

t1

∫〈b, rσµs − i〉dµs ds+

∫ t2

t1

(H(σ)−H(µs)) ds

(3.4.26)holds for any σ ∈ D(H) and t2 > t1 > 0 if b satisfies a condition which is impliedby (3.2.3). Recall that for measures µ ∈ Prt (Rd) and σ ∈ P2(Rd), rσµ denotesthe unique optimal transport map between µ and σ. A simple application of theIto formula yields that for any initial measure µ0 ∈ P2(Rd) the semigroup patht 7→ Rtµ0 is a distributional solution of (3.1.1), thus Proposition 3.4.2 implies that(3.4.26) holds for µt := Rtµ0 for any µ0 ∈ D(H) and 0 < t1 < t2. We are going touse this fact in the proof of our next proposition. Observe that Proposition 3.4.3implies that the product formula in Theorem 3.3.12 defines distributional solutionsof the non-symmetric Fokker-Planck equation (3.1.1), which enjoys similar pathregularity properties as the gradient flow semigroup associated to the functional

97



H (i.e. the semigroup of the solutions of the heat equation). For instance, evenif µ0 is not an absolutely continuous measure, for each initial point µ0 ∈ P2(Rd),and for each ε > 0, the curve ε 6 s 7→ Rsµ0 is a Wasserstein solution in the senseof [79] Definition 5.1, and moreover it is Lipschitz on [ε,+∞).

Proposition 3.4.3. Assume that b satisfies (3.2.1) and (3.2.2). Then for anyµ0 ∈ P2(Rd) the curve t 7→ µt = Rtµ0 is Lipschitz on [ε,+∞) for any ε > 0, and|∂H|(µt) < +∞ for each t > 0.

Proof. Fix an arbitrary µ0 ∈ P2(Rd). By Proposition 3.4.2 it is enough to showthat H(µt) < +∞ for t > 0. Indeed once we prove this claim, we will know thatthe assertions of the proposition hold for each curve s 7→ µt+s and arbitrary t > 0,which completes the proof. To show the claim, pick a sequence µ0,k → µ0 suchthat for all k ∈ N |∂H|(µ0,k) < +∞—such a sequence exists due to (3.2.35) and(3.2.42). Denote further µt,k := Rtµ0,k for k ∈ N and t > 0. By (3.4.3)

H(µt,k) 6 H(µ0,k) + tl (3.4.27)

holds for t > 0, hence for t > s > 0 and k ∈ N

H(µt,k) 6 H(µs,k) + (t− s)l (3.4.28)

and integrating this inequality gives

tH(µt,k) 6∫ t

0

H(µs,k) ds+t2l

2. (3.4.29)

Combining (3.4.29) with (3.4.26) and rearranging terms gives that

H(µt,k) 6tl

2+H(σ) +

12t

∫ t

0

∫Rd〈b, rσµs,k − i〉dµs,k ds

+12tW 2

2 (µ0,k, σ)− 12tW 2

2 (µt,k, σ)

6tl

2+ (H(σ) +

12tW 2

2 (µ0,k, σ))

+14t

∫ t

0

(∫Rd|b(x)|2 dµs,k(x) +W 2

2 (µs,k, σ))

ds− 12tW 2

2 (µt,k, σ)

(3.4.30)for any σ ∈ D(H) 6= ∅, where the second inequality in (3.4.30) follows by theCauchy-Schwarz inequality. Recall (3.2.37) and (3.2.38), take ϕ = H there, andchoose σk := Itµ0,k in (3.4.30) to obtain

H(µt,k) 6 Ht(µ0,k) +tl

2+

12t

∫ t

0

∫Rd|b(x)|2 dµs,k(x) ds

+12t

∫ t

0

W 22 (µs,k,Jtµ0,k) ds− 1

2tW 2

2 (µt,k,Jtµ0,k).(3.4.31)

98

Section 3.4

Well now for a fixed t > 0 both mappings It : D(|∂H|) = P2(Rd)→ P2(Rd) andHt : P2(Rd) → R are continuous (see Section 3.2). Furthermore for each k ∈ Nand 0 6 s 6 t the contraction property of (Rt)t>0 implies that W2(µs,k, µs) 6W2(µ0,k, µ0) → 0 for each s > 0 and since |b|2 is a continuous function with atmost quadratic growth we have that

∫Rd |b(x)|2 dµs,k(x) n→+∞−→

∫Rd |b(x)|2 dµs for

each s > 0. Finally (3.3.28) and Theorem 3.3.12 imply that µs : s ∈ [0, t]is a bounded set. In light of the bounded convergence theorem and lower semi-continuity of H we estimate

H(µt) 6 lim infk→∞

H(µt,k) 6 Ht(µ0) +tl

2− 1

2tW 2

2 (µt,Jtµ0)

+12t

∫ t

0

∫Rd|b(x)|2 dµs ds+

∫ t

0

W 22 (µs,Jtµ0) ds < +∞

(3.4.32)

We have completed the proof now.

Remark 3.4.4. In the above proof we could have argued without invokingHt(µ0).However the estimate (3.4.32) we obtained in this way seems to be better thanthe estimate we would obtain by fixing one particular σ ∈ D(H) and subsequentlyarguing finiteness of H(µt).

To conclude this section, let us summarise the properties of the semigroup(Rt)t>0 of solutions of equation (3.1.1) in a theorem. It is actually evident thatthis semigroup has the same path and contraction properties as gradient flowson P2(Rd) do. However we do not have the resolvent operators. In order tostate and prove this theorem we need two propositions first. In the first of thesetwo propositions we argue a priori uniqueness of solutions of (3.1.1) within theclass of locally absolutely continuous curves in P2(Rd). A uniqueness result forequations like (3.1.1) is not new by any means (see for instance [13] and [80]).We will give a (different) proof in order for this monograph to be self contained.The second proposition is a general fact about contraction semigroups on metricspaces, which should not come as a surprise. For example a related claim in thesetting of gradient flows on metric spaces is proven in [5] Theorem 2.4.15 (i). Weprovide a proof of this fact in our setting.

Proposition 3.4.5. Suppose that b : Rd → Rd satisfies (3.2.2) and that for someα ∈ R

〈b(x)− b(y), x− y〉 > 2α|x− y|2 ∀x, y ∈ Rd. (3.4.33)

Consider two continuous curves 0 6 t 7→ µjt ∈ P2(Rd), j = 1, 2, whose re-strictions to (0,+∞) are distributional solutions of equation (3.1.1) such thatµjt = ρt · Ld for L1-a.e. t > 0. Suppose moreover that both solutions are ofclass AC2

loc((0,+∞);P2(Rd)) and continuous up to 0, and suppose moreover that

the vector fields (t, x) 7→ vjt (x) := ∇ρjt(x)

ρjt(x)+ b(x) are defined for L1-a.e. t > 0 and

for each such t > 0 satisfy∫ T

0

|vjt |2L2(µt;Rd) dt < +∞ ∀T > 0 (3.4.34)

99



for j = 1, 2. Then the following estimate holds

W2(µ1t , µ

2t ) 6 e−tαW2(µ1

0, µ20) ∀t > 0 (3.4.35)

Proof. Recall the regular tangent spaces defined in (3.2.32). Since b satisfies thelinear growth condition (3.2.3) and in particular b ∈ L2(µjt ; Rd) holds for eacht > 0 and for j = 1, 2, [5] Theorem 10.4.6, [5] Theorem 11.1.3 and the assumption(3.4.34) imply that

∇ρjtρjt∈ Tanµjt (P2(Rd)) L1-a.e., t > 0, (3.4.36)

for j = 1, 2. For t > 0 where vjt is defined, define

vjt := −

(Pµjt

b+∇ρjtρjt

)j = 1, 2 (3.4.37)

where Pµjt denotes the orthogonal projection in L2(µjt ; Rd) onto its closed linear

subspace Tanµjt (P2(Rd)). Observe that (3.4.36) implies that vjt ∈ TanµjtP2(Rd)if defined, thus in particular (t, x) 7→ vjt is the tangent vector field of the curvet 7→ µjt for j = 1, 2 (see [5] Theorem 8.3.1).

Fix T > 0 and define function

δ(t, s) := W 22 (µ1

t , µ2s), s, t ∈ (0, T ). (3.4.38)

For each t, s ∈ (0, T ) fix an optimal transport plan γt,s ∈ Γ0(µ1t , µ

2s). Since vjt is

the tangent vector field, [5] Theorem 8.4.7 , [5] Theorem 8.3.1, and the Cauchy-Schwarz inequality give

ddtδ(t, s) =

∫Rd〈v1t (x1), x2 − x1〉dγt,s(x1, x2) 6 |µ1|(t)W2(µ1

t , µ2s) (3.4.39)

and

ddsδ(t, s) =

∫Rd〈v2s(x2), x1 − x2〉dγt,s(x1, x2) 6 |µ2|(s)W2(µ1

t , µ2s). (3.4.40)

Next we define the function

v(t) := supt,s∈(0,T )

W2(µ1t , µ

2t )(|µ1|(t) + |µ2|(t)) (3.4.41)

so that due to our local absolute continuity assumptions and the linear growth of bwe can conclude that the function δ(t, s) satisfies the conditions in [5] Lemma 4.3.4.Henceforth (0, T ) 3 t 7→ δ(t, t) = W 2

2 (µ1t , µ

2t ) is a locally absolutely continuous

function and moreover

ddtδ(t, t) 6 lim sup

h↓0

δ(t, t)− δ(t− h, t)h

+ lim suph↓0

δ(t, t+ h)− δ(t, t)h

(3.4.42)

100

Section 3.4

holds for L1-a.e. t ∈ (0, T ). Thus for L1-a.e. t > 0 we have that

ddtW 2

2 (µ1t , µ

2t ) 6

∫Rd〈v1t (x1)− v2

t (x2), x2 − x1〉dγt,t(x1, x2)

=∫

Rd〈v1t (x1)− v2

t (x2), x2 − x1〉dγt,t(x1, x2) L1-a.e. t ∈ (0, T )

(3.4.43)where the equality in (3.4.43) follows by [5] Theorem 8.5.1. In light of monotonicityof the Wasserstein-2 subdifferential of H (see [5] (10.1.8)), (3.4.33) and (3.4.43) weobtain

ddtW 2

2 (µ1t , µ

2t ) 6

∫Rd〈b(x1)− b(x2), x2 − x1〉dγt,t(x1, x2)

6∫

Rd−2α|x1 − x2|2 dγt,t(x1, x2) = −2αW 2

2 (µ1t , µ

2t )

(3.4.44)

for L1-a.e. t ∈ (0, T ). Since both curves are assumed continuous up to 0 and T > 0was arbitrary we conclude (3.4.35).

Proposition 3.4.6. Let X be a metric space and let (Ft)t>0 be a contractionsemigroup on X with paths of class AC1((0, T );X) for each 0 < T <∞. Then foreach x ∈ X and for each t > 0 the L1-a.e. defined metric derivative |u|(t) of thecurve u(t) = Ftx has a (unique non-increasing) right continuous version t 7→ g(t).Moreover, t 7→ u(t) is right metrically differentiable at each t > 0 and

limh↓0

d(u(t+ h), u(t))h

= g(t). (3.4.45)

Proof. Fix x ∈ X. Since by assumption the curve u(t) := Ftx is absolutelycontinuous on each compact interval its metric derivative |u|(t) is L1 a.e. definedon [0,+∞). Due to the contraction assumption if 0 < 0t1 6 t2 <∞ are two timepoints where the metric derivative is defined we have for each h > 0

d(u(t1 + h), u(t1))h

>d(u(t2 + h), u(t2))

h, (3.4.46)

hence t 7→ |u|(t) is L1 a.e. equal to a non-increasing function. This implies thatthere is a unique right continuous non-increasing function t 7→ g(t) on [0,+∞)such that g is L1-a.e. equal to the metric derivative of t 7→ u(t).

Next for each t > 0 and for each h > 0 we have

d(u(t+ h), u(t))h

6∫ t+h

t

g(s)ds,

hence

lim suph↓0

d(u(t+ h), u(t))h

6 g(t). (3.4.47)

101



On the other hand if t2 > t is such that (3.4.45) holds with t replaced by t2,(3.4.46) yields

lim infh↓0

d(u(t+ h), u(t))h

> lim infh↓0

d(u(t2 + h), u(t2))h

= g(t2) ↑ g(t) (3.4.48)

for t2 ↓ t since g is a right continuous function. The pointwise right metric differ-entiability claim now follows by (3.4.47) and (3.4.48).

At last we are going to state and prove our final theorem in this section. Thereader may observe that our Theorem 3.4.7 and Theoreme 3.1 in [17] are analogous,except that we did not define the resolvents.

Theorem 3.4.7. Let b satisfy (3.2.1) and (3.2.2). Then for each µ0 ∈ P2(Rd)the curve t 7→ µt := Rtµ0 is the unique curve with values in P2(Rd) such that

∂tρt = ∇ · ((∇ρtρt

+ b)µt) = ∆ρt +∇ · (bρt) (3.4.49)

holds in D′((0,+∞)× Rd) and such that the following two properties hold

1 µt ∈ D(|∂H|) for each t > 0

2 t 7→ µt is of class AC2loc((0,+∞);P2(Rd)) and continuous up to 0.

In addition we have

3 t 7→ µt is Lipschitz continuous on [ε,+∞) for each ε > 0 and denotingρt := dµt

dLd we have that∫ T

0

|∇ρtρt

+ b|2L2(µt;Rd) dt < +∞ (3.4.50)

for each T > 0. If µ0 ∈ D(|∂H|) then 0 6 t 7→ µt is Lipschitz and also

|µ|(t) 6 |∇ρ0

ρ0|L2(µ0;Rd) + |b|L2(µ0;Rd) ∀t > 0 (3.4.51)

4 t 7→ µ(t) is right metrically differentiable at each t > 0 and also at t = 0 ifµ0 ∈ D(|∂H|). Its metric derivative has a right continuous non-increasingversion .

5 If t 7→ µt is another solution of (3.1.1) with properties 1, 2 and 3 starting atµ0, then

W2(µt, µt) 6 W2(µ0, µ0) ∀t > 0. (3.4.52)

102

Section 3.5

Proof. Uniqueness of solutions of (3.4.49) subject to 1 and 2 follows directly fromProposition 3.4.5. 1 follows by Proposition 3.4.3. In order to prove continuity upto 0 of t 7→ Rtµ0 claimed in 2 notice first that by Proposition 3.4.3 if ν ∈ D(|∂H|)the curve 0 6 t 7→ Rtν is Lipschitz. Choose ε > 0 and pick νε ∈ D(|∂H|)such that W2(µ0, ν

ε) < ε. Moreover let t0 > 0 be such that for t < t0 we haveW2(Rtνε, νε) < ε. Then by the contraction property of (Rt)t>0 we estimate

W2(Rtµ0, µ0) 6 W2(Rtµ0, Rtνε) +W2(Rtνε, νε) +W2(νε, µ0) < 3ε

for all t < t0. Hence t 7→ Rtµ0 is continuous up to 0 indeed. The remainder of 2 isproved in Proposition 3.4.3. Next let us show (3.4.51) with the aid of Proposition3.4.6. By Theorem 3.3.12 we have that

1tW2(µt, µ0) = lim

n→+∞

1tW2((R2,1

tn

)nµ0, µ0), (3.4.53)

therefore for each t > 0 we estimate

lim supn→+∞

1tW2((R2,1

tn

)nµ0, µ0) 6 lim supn→+∞

n−1∑k=0

W2((R2,1tn

)k+1µ0, (2,1tn

)kµ0)

t

6 lim sup

n→+∞

W2(R2,1tn

µ0, µ0)

t/n

6 lim supn→+∞

W2(R1t/nµ0, µ0)

t/n+ lim sup

n→+∞

W2(R2t/nµ0, µ0)

t/n

6 |b|L2(µ0;Rd) + |∇ρ0

ρ0|L2(µ0;Rd),

(3.4.54)where the latter inequality follows by (3.2.44), Theorem 3.3.8, and Lemma 3.3.7.Taking lim supt↓0 in (3.4.53) and (3.4.54) gives (3.4.51).

4 follows by Proposition 3.4.6 and 5 follows by Proposition 3.4.5.

3.5 Some remarks about the invariant measure—the symmetric versus the non-symmetric case

It is well known that if b satisfies condition (3.4.33) then SDE (3.2.7) has a uniqueinvariant measure. And if moreover b = ∇V where V : Rd → (−∞,+∞] is aconvex lower semi-continuous function then the semigroup (Rt)t>0 is a gradientflow semigroup associated to the functional

H(µ|µ∞) =

∫Rd

dµdµ∞

log dµdµ∞

dx if µ µ∞

+∞ otherwise,(3.5.1)

where µ∞ denotes the unique invariant measure. One also easily verifies thatan equivalent definition of the functional µ 7→ H(µ|µ∞) is given by H(µ|µ∞) =

103



∫Rd log( dµ

dLd ) dLd+∫

Rd V dµ if µ Ld andH(µ|µ∞) = +∞ otherwise. In that casethe semigroup of solutions (Rt)t>0 can be constructed as the limit of the iterationsof the resolvents (associated to H(·|µ∞)), i.e. by the exponential formula (3.2.36).In particular the invariant measure contains the full information about the flow,since the resolvents are fully determined by the invariant measure.

One may wonder whether in case where b is not a gradient, and the equation(3.1.1) has a unique invariant state µ∞, the semigroup (Rt)t of solutions of thenon-symmetric Fokker-Planck equation can be recovered from µ∞. Intuitivelyspeaking one might guess that the answer is negative. Our contributions in thissection are Proposition 3.5.4 and the discussion below it, which provide a rigorousproof that the answer is indeed negative, in the case where b is linear. For theproof, we will consider a linear positive mapping b, and then argue by linking theorthogonal projections Pµ∞ : P2(Rd) → Tanµ∞P2(Rd) with certain orthogonalprojections in the space of matrices. Here we endow the space of d × d matriceswith the inner product associated to the symmetric positive matrix Q := ∇ρ∞

ρ∞,

ρ∞ := dµ∞dLd (details can be found below).

For the sake of completeness of our arguments we will start by reproducingthe existence and uniqueness of the invariant measure for (3.1.1) by means ofestimate (3.4.35). We will add some consequences concerning the invariant mea-sure. These facts are harder to prove by more traditional techniques such as theKrylov-Bogoliubov method. Similar facts as in the following proposition have beenobserved in [80].

Proposition 3.5.1. Suppose b : Rd → Rd satisfies (3.2.2) and (3.4.33) withα > 0. Then there is a unique constant distributional solution 0 6 t 7→ µt = µ∞ ∈P2(Rd) of equation (3.1.1) and we have µ∞ ∈ D(|∂H|). Moreover a measureµ ∈ P2(Rd) satisfies µ = µ∞ if and only if µ = ρ · Ld for some ρ ∈W 1,1

loc (Rd) suchthat ∫

Rd〈∇ϕ, ∇ρ

ρ+ b〉ρdx = 0 ∀ϕ ∈ C∞c (Rd) (3.5.2)

and for any µ0 ∈ P2(Rd) we have the following estimate

W2(Rtµ0, µ∞) 6 e−tαW2(µ0, µ∞) ∀t > 0. (3.5.3)

Proof. First of all by Theorem 3.4.7 the solutions of (3.1.1) are precisely the pathsof (Rt)t>0. But then since for each t > 0 Rt is a strict contraction, by the Banachfixed point theorem this mapping has a unique fixed point, thus there can notbe more than one invariant measure for the whole semigroup either. Denote theunique fixed point ofRt by µ∞,t for t > 0. Let us show that for s, t > 0 µs,∞ = µt,∞holds. Well if s = kt for k ∈ N then Rsµt,∞ = (Rt)kµt,∞ = µt,∞, hence as Rs canhave only one invariant measure, we must have µs,∞ = µt,∞, and by the symmetryalso if t = ks for k ∈ N. This implies that if t

s = pq for any p, q ∈ N we have that

µs,∞ = µps,∞ = µqt,∞ = µt,∞ (3.5.4)

as well. Finally since all paths of (Rt)t>0 are continuous by Theorem 3.4.7, (3.5.4)follows for all s, t > 0.

104

Section 3.5

To prove the ‘only if’ part of the stated equivalence observe that by Proposition3.4.3 the invariant measure µ = µ∞ must satisfy µ∞ = Rtµ∞ ∈ D(|∂H|) for eacht > 0. Therefore in light of (3.2.43) we have that µ∞ = ρ∞ · Ld, ρ∞ ∈ W 1,1

loc (Rd)and ∇ρ∞ρ∞ ∈ L2(µ∞; R), and (3.4.49) implies that (3.5.2) must hold. Conversely ifP2(Rd) 3 µ := ρ · Ld with ρ ∈W 1,1

loc (Rd) satisfies (3.5.2), then clearly the constantcurve t 7→ µt ≡ µ is a solution of (3.1.1) and µ is the invariant measure for equation(3.1.1). Estimate (3.5.3) follows by Theorem 3.4.7 and (3.4.35).

Remark 3.5.2. Notice that Proposition 3.5.1 also gives existence and uniquenessin W 1,1

loc (Rd) of the elliptic problem (3.5.2). In the remainder of this section wewill denote the invariant measure of (3.1.1) by µ∞ = ρ∞ · Ld.

Throughout the remainder of this section we assume that b = A ∈ Rd2= Rd×d

is strictly positive definite but non-symmetric matrix. Recall that a positive matrixis a gradient if and only if it is symmetric and if it is symmetric then

A = ∇ψ, ψ(x) :=12〈Ax, x〉 (3.5.5)

and ψ is a convex C∞ function. Such a matrix A always satisfies (3.4.33) for somepositive α since α := inf06=x∈Rd

〈Ax,x〉|x|2 > 0 by compactness of the unit sphere in

Rd. In this case the invariant measure µ∞ is described as follows (c.f. [32] Section(6.2.1)). For t > 0 we define St := exp(−tA) and

Qt :=∫ t

0

SsS∗sds, (3.5.6)

which is a symmetric strictly positive definite matrix. Then for any µ0 ∈ P2(Rd)we have that Rtµ0 = µ0 ∗ N (0, Qt) where N (0, Qt) is the d-dimensional Gaussianmeasure with mean 0 and covariance matrix Qt. Moreover the invariant measureis µ∞ = N (0, Q∞) where

Q∞ =∫ ∞

0

StS∗t dt (3.5.7)

and its density is given by

ρ∞(x) :=e−〈Q

−1∞ x,x〉/2√

(2π)ddetQ∞. (3.5.8)

Proposition 3.5.3. Let A ∈ Rd2be a strictly positive definite matrix and let Q

be a symmetric strictly positive definite matrix. Then for the Gaussian measureN (0, Q) the following three conditions are equivalent

1 N (0, Q) is the invariant measure for equation (3.1.1) with b := A

2 The following identity holds

PN (0,Q)A = Q−1 (3.5.9)

105



3 Denoting tr : Rd2 → R to be the trace functional we have that

trA− trQ−1 = 0 and 〈Q−1x, (A−Q−1)x〉 = 0 ∀x ∈ Rd. (3.5.10)

Proof. Denote ρ(x) := e−〈Q−1x,x〉/2√

(2π)ddetQto be the density function of N (0, Q) and

observe that ∇ρρ = −Q−1 ∈ L2(N (0, Q); Rd) holds. Notice furthermore thatQ−1 is also a strictly positive definite matrix thus by (3.5.5) a gradient. Anapproximation argument easily yields that

−Q−1 ∈ TanN (0,Q)P2(Rd). (3.5.11)

Let us first prove the equivalence between items 1 and 2. Well by Proposition3.5.1 N (0, Q) is invariant if and only if

0 =∫〈∇ϕ(x), (−Q−1 +A)x〉 e

−〈Q−1x,x〉/2√(2π)ddet Q

dx (3.5.12)

for all ϕ ∈ C∞c (Rd), and in light of (3.5.11) this is the case if and only if (3.5.9)holds.

Next we prove the equivalence between items 2 and 3. Well by integration byparts we deduce that (3.5.12) holds if and only if

0 =∫ϕ(x)(trA− trQ−1)ρ(x)dx−

∫ϕ(x)〈Q−1x, (A−Q−1)x〉ρ(x)dx (3.5.13)

holds for all ϕ ∈ C∞c (Rd), which is the case if and only if

〈Q−1x, (A−Q−1)x〉 = trA− trQ−1 ∀x ∈ Rd. (3.5.14)

Evaluating these expressions for x = cy for c ∈ R we conclude that (3.5.14) holdsif and only if

trA = trQ−1, and 〈Q−1x, (A−Q−1)x〉 = 0 ∀x ∈ Rd, (3.5.15)

which is the condition stated in item 3.

Next denote Vd ⊂ Rd2to be the subspace of symmetric matrices and let C ∈ Vd

be a symmetric strictly positive definite matrix. Then one can define the 〈·, ·〉Cinner product on the space of d−dimensional matrices Rd2

by

〈U, V 〉C := trC−1UV for U, V ∈ Rd2

(3.5.16)

and also the C inner product on Rd by

〈x, y〉C := 〈C−1x, y〉 for x, y ∈ Rd (3.5.17)

106

Section 3.5

Further PVd〈·,·〉C will denote the orthogonal projection onto the linear subspace Vd

of Rd2with respect the the inner product 〈·, ·〉C . If C is the identity matrix then

a direct computation yields that the orthogonal complement of Vd in Rd2with

respect to 〈., .〉C is the linear subspace of skew-symmetric matrices in Rd2.

We also recall the usual notation U t for the transpose of a matrix U ∈ Rd2.

In our final proposition of this chapter we give another characterisation of theinvariant measure when the drift is a matrix. This characterisation will enable usto see that the same invariant measure may be shared by many different Fokker-Planck equations.

Proposition 3.5.4. Let A ∈ Rd2be a strictly positive definite matrix and let Q be

a symmetric strictly positive definite matrix. Then the Gaussian measure N (0, Q)is invariant for (3.1.1) if and only if

Q−1 = PVd〈·,·〉QA. (3.5.18)

Proof. Let Q := PVd〈·,·〉QA. By definition this is equivalent with

0 = trQ−1(A− Q)D∗ = trQ−1(A− Q)D (3.5.19)

for each D ∈ Vd. Moreover (3.5.19) is equivalent with Q−1(A − Q) being anti-symmetric i.e.

Q−1(A− Q) = −(At − (Q)t)(Q−1)t = (At − Q)Q−1. (3.5.20)

Define ψ(x) := 〈Q−1x, (A− Q)x〉 = 〈x,Q−1(A− Q)x〉 for x ∈ Rd and observethat

∇ψ = Q−1(A− Q) + (Q−1(A− Q))t = Q−1(A− Q) + (At − Q)Q−1 (3.5.21)

and that ψ(0) = 0.Thus if (3.5.20) holds then

ψ(x) = 〈Q−1x, (A− Q)x〉 = 0 ∀x ∈ Rd (3.5.22)

and taking D := Q ∈ Vd in (3.5.19) implies that

0 = trQ−1(A− Q)Q = tr(A− Q) (3.5.23)

must hold as well. Conversely if (3.5.22) holds then ∇ψ ≡ 0 which implies (3.5.20).Now suppose that Q = Q−1 holds. Then, as we showed above, (3.5.22) and

(3.5.23) hold, hence by Proposition 3.5.3, N(0, Q) is the invariant measure. Con-versely suppose that N (0, Q) is the invariant measure. Then by the Proposition3.5.3 (3.5.10) holds, hence (3.5.20) holds, too, which is equivalent to (3.5.18). Theproof is now complete.

107



Now we are able to explain why the invariant measure µ∞ = N (0, Q∞) doesnot contain all the information about the flow (Rt)t>0 if A is not a gradient. Setα := infx∈Rd〈Ax, x〉 > 0, and pick any matrix D ∈ Vd⊥—the 〈·, ·〉Q∞ orthogonalcomplement of Vd, such that |D|Rd2 < α (|.|Rd2 denotes the Euclidean norm onRd2

). Then for any 0 6= x ∈ Rd we have |〈(A−D)x, x〉| > α|x|2 − |D|Rd2 |x|2 > 0,i.e. the matrix A1 := A−D is also a strictly positive definite. Since by the choiceof D we clearly have that PVd

〈·,·〉Q∞A = PVd

〈·,·〉Q∞A1, the previous proposition yields

that N (0, Q∞) is also the invariant measure for the equation (3.1.1) with b := A1.Thus there are many equations of the Fokker-Planck type with the same invariantmeasure N (0, Q∞).

108

Chapter 4

Maximal Monotoneoperators in generalisedsense on the Wassersteinspace P2(Rd)

4.1 Introduction

During the 1960’s and 1970’s, convex functionals and their Frechet subdifferentials,maximal monotone operators on Hilbert spaces, and the induced semigroups havebeen studied extensively. The emerged theory is a natural generalisation of thetheory of the linear C0-semigroups, as we can see from the non-linear version ofthe Hille-Yosida theorem (see [17] Theoreme 4.1) The subdifferential of a convexand lower semi-continuous functional on a Hilbert space is a maximal monotoneoperator, and conversely, we have Rockafeller’s theorem (see [17] Theoreme 2.5),which states that any cyclically monotone operator on a Hilbert space, is a subsetof the subdifferential of a convex, lower semi-continuous functional. Therefore, thetheory of maximal monotone operators on Hilbert spaces is more general than thetheory of gradient flows.

During the past decade, we have witnessed a major development of the theoryof gradient flows in various metric space settings, and in particular in the settingof (P2(Rd),W2) (see [54], [85], [5], [22], [7], [40], [96], [83], among many othermonographs). The Wasserstein-2 counterpart of the theory of gradient flows onHilbert spaces is now well established, and there are many analogies between thetwo settings. Besides being rigorous and elegant abstract mathematics, the theoryof gradient flows on P2(Rd) has also found many applications. For example, variousclasses of PDE’s can be treated in such way, such as the Fokker-Planck equations(see [54]), the porous medium equations (see [85]), and various other equations

109

CHAPTER 4: Maximal Monotone operators in generalised sense on the Wasserstein

space P2(Rd)

(see [5]).Furthermore, various infinite dimensional formulations of Wasserstein-2 gra-

dient flows have been considered, i.e. the gradient flows on (P2(H),W2), whereH is a separable Hilbert space (see [5], [7], and [6]), and also the gradient flowon Cameron-Martin spaces (see [40] and [70] for the Fokker-Planck equation inthis context, and [41] for a treatment of the Monge-Kantorovich problem in thiscontext). It is worth mentioning that the theory of gradient flows on Wassersteinspaces is an integral part of the optimal transportation theory, which has otherapplications as well. For instance, applications in image processing or in medicalpractice in establishing diagnoses of certain lung illnesses (see [77]).

An important generalisation of gradient flows in Hilbert spaces are abstractCauchy problems associated to maximal monotone operators. If A is a maximalmonotone operator on a Hilbert space H, the Cauchy problem associated to Areads: find a curve u : [0,+∞) → H which is absolutely continuous on [0, T ] foreach T > 0 such that

d

dtu(t) ∈ −Au(t) L1-a.e. t > 0,

u(0) := u0 ∈ D(A).(4.1.1)

An natural question to consider is whether this theory has a counterpart in thetheory of the Wasserstein-2 space P2(Rd). The first point required in this regardis finding a suitable definition of maximal monotone operators on P2(Rd). Thisdefinition should be such that an associated Cauchy problem can be posed andsolved, and such that it includes the subdifferentials of the functionals that areconvex along generalised geodesics from the theory of [5]. This chapter presentssuch a definition of maximal monotone operator and develops the correspondingtheory.

In Section 4.2 we recall basic facts from the theory of maximal monotoneoperators on Hilbert spaces. In Section 4.3 we recall some properties of the subd-ifferentials of geodesically convex functionals on P2(Rd).

In Section 4.4 we start the exposition of the new theory by presenting ourdefinition of maximal monotone operators on P2(Rd) and we associate an abstractCauchy problem to such operators.

Compatibility of our definition with the existing theory of subdifferentials offunctionals is discussed in Section 4.5. We define a suitable subset of subdiffer-entials of generalised convex functionals on P2(Rd), which are maximal monotoneoperators according to our definition. Consequently our theory extends the theoryof regular generalised convex functionals presented in [5].

Level sets of convex functionals on Hilbert spaces are convex. Section 4.6discusses an analogous notion of convexity related to generalised convex functionalson P2(Rd). This notion and its properties are closely related to our maximalmonotone operators.

In the classical theory on Hilbert spaces resolvents associated to maximal mono-tone operators play central role in the construction of the solutions of the associ-ated Cauchy problem and in proving the properties of the induced flow. Proofs of

110

Section 4.1

several properties of the resolvents associated to our maximal monotone operatorsare presented in Section 4.7.

Section 4.8 is devoted to the main result of our theory, namely the solutionsof the Cauchy problem. We establish existence, uniqueness, the semi-contractionproperty of the induced semigroup, the locally Lipschitz property of its paths, andseveral other properties analogous to the Hilbert space case.

To conclude this introduction, let us briefly discuss the assumptions which weare going to make in order to prove the results of this chapter.

In the monographs [72] and [5]1, and in Chapter II of this thesis, a large partof the basic theory of gradient flows on Hilbert spaces has been generalized tothe setting of complete CAT(0) spaces. These results show that such a settingis a natural generalization of the theory of gradient flows on Hilbert spaces, andthe lack of the inner product structure is overcome. We recall from Chapter IIthat (X, d) is a CAT(0) space if it is a complete geodesic space of non-positivecurvature in the sense of Alexandrov, where the latter means that for any tripleof points v0, v1, w ∈ X and any geodesic γ : [0, 1] → X that connects v0 with v1

the following estimate holds:

d2(w, γ(t)) 6 (1− t)d2(w, v0) + td2(w, v1)− t(1− t)d2(v0, v1) ∀t ∈ [0, 1] (4.1.2)

Moreover in [5] the authors explain that the main idea in reproducing the Hilber-tian gradient flows theory in a setting of a general complete metric space X isthe assumption of existence of generalized geodesics. That is for each triple ofpoints v0, v1, w ∈ X they assume existence of a curve γ : [0, 1] → X , such thatγ(0) = v0, γ(1) = v1 and

d2(w, γ(t)) 6 (1− t)d2(w, v0) + td2(w, v1)− t(1− t)d(v0, v1), (4.1.3)

while the considered proper, lower semi-continuous functional ϕ : X → (−∞,+∞]is λ-convex along γ (for some λ ∈ R)2, i.e.

ϕ(g(t)) 6 (1− t)ϕ(v0) + tϕ(v1)− λ

2t(1− t)d2(v0, v1). (4.1.4)

It is worth mentioning that there is no approximation theory for gradient flows,and in particular there is no product formula, at a general level in this setting sofar, in spite of attempts of the author of this thesis to provide one.

The space (P2(Rd),W2) is positively curved in the sense of Alexandrov, i.e. wehave > in (4.1.2) instead of 6 as it is proven in [5] Theorem 7.3.2 (equivalently thecomparison triangle of each triangle in P2(Rd) is thicker, see Chapter II). However,for any triple of points µ0, µ1, σ ∈ P2(Rd) and any 3-plan γ ∈ P2(R3d) such that

π0,1# γ ∈ Γo(σ, µ0), π0,2

# γ ∈ Γo(σ, µ1) (4.1.5)

1 Since CAT(0) spaces satisfy the generalized convexity assumption in [5] (see Assumption4.0.1 there), the results presented in Part I of this monograph apply to CAT(0) spaces.

2There are other assumptions and techniques to construct gradient flows, such as that thefunctional has compact level sets of that it is locally Lipschitz and X has a lower curvature boundin the sense of Alexandrov see [86] and [69]. However we will not employ these ideas in designingthe theory of our Wasserstein-2 maximal monotone operators.

111


space P2(Rd)

the curve defined byγ(t) := ((1− t)π1 + tπ2)#γ (4.1.6)

satisfies estimate (4.1.3), with v0 := π1γ, v1 := π2γ (see (4.4.9) for the definition ofthe coordinate projections π1 and π2). If a functional defined on (P2(Rd),W2) isλ-convex along such curves (λ ∈ R), one can apply the theory developed in the firstpart of [5]. The impetus for investigations in this chapter is the idea that we shouldbe able to give a genaralisation of the theory of gradient flows on (P2(Rd),W2), byexploiting the generalized convexity of W 2

2 , in order to compensate for the positivecurvature of (P2(Rd),W2).

4.2 Maximal monotone operators on Hilbertspaces

Let us briefly recall the main features of maximal monotone operators on a realHilbert space H. In this subsection H denotes a fixed Hilbert space, 〈., .〉 denotesthe inner product on H, and |.| denotes the induced norm. A subset A ⊂ H ×His monotone if we have

〈y1 − y2, x1 − x2〉 > 0 ∀[x1, y1], [x2, y2] ∈ A, (4.2.1)

and it is maximal monotone if there exists no monotone subset of H × H thatcontains A as a proper subset. A subset A ⊂ H × H is sometimes viewed as a(multivalued) operator. For an operator A ⊂ H × H its domain and image aredefined by

D(A) := x ∈ H : ∃y ∈ H such that [x, y] ∈ A,R(A) := y ∈ H : ∃x ∈ H such that [x, y] ∈ A.

(4.2.2)

We further denote

Ax := y ∈ H : [x, y] ∈ A, x ∈ D(A). (4.2.3)

It is not hard to show that if A is a monotone operator then the resolvent operators(I+hA)−1 are single valued where defined, and maximal monotone operators havethree equivalent characterizations which we state in the following proposition.

Proposition 4.2.1. Let H be a Hilbert space and let A ⊂ H × H. Then thefollowing properties are equivalent:

(i) A is maximal monotone

(ii) A is monotone and R(I +A) = H

(iii) For each h > 0 (I + λA)−1 is a contraction defined everywhere on H.

Proof. See [17] Proposition 2.2.

112

Section 4.2

The remainder of this section is devoted to the basic properties of maximalmonotone operators on Hilbert spaces. These properties will help us to definea reasonable analogue of such objects in the setting of P2(Rd). Throughout theremainder of this section A ⊂ H ×H denotes a maximal monotone operator on aHilbert space H.

The resolvents of A are defined by

Jhx := (I + hA)−1x, h > 0, x ∈ H. (4.2.4)

Thus for x ∈ D(A) and h > 0 the identity x1 = Jhx means that [x1, y1] ∈ A isthe unique pair of points in A such that x = x1 + hy1. The existence and theuniqueness of solutions of the problem (4.2.4) follows by Proposition 4.2.1.

Proposition 4.2.2. Let A ⊂ H ×H be a maximal monotone operator. Then thefollowing resolvent identity holds:

Jh1x = Jh2(h2

h1x+ (1− h2

h1)Jh1x) ∀x ∈ H, ∀ 0 < h2 6 h1. (4.2.5)

Proof. Fix 0 < h2 6 h1 and x ∈ H. By (4.2.4) there is a unique ξ1 ∈ AJh1x suchthat x = Jh1x+h1ξ holds. Multiplying both sides by h2

h1, and adding (1− h2

h1)Jh1x

to both sides givesh2

h1x+ (1− h2

h1)Jh1x = Jh1x+ h2ξ. (4.2.6)

Since [Jh1x, ξ] ∈ A and since the resolvents are single valued, (4.2.5) follows.

Theorem 4.2.3. For each x ∈ D(A) we have that Ax is a closed convex subset ofH. The set D(A) is convex, and denoting P

D(A)to be the nearest point projection

onto D(A), we havelimh↓0

Jhx = PD(A)

x ∀x ∈ H. (4.2.7)

Proof. See [17] Theoreme 2.2.

By the above theorem, for each x ∈ D(A) there is a unique element of minimalnorm in Ax and we denote

Aox := argmin|y|; y ∈ Ax for x ∈ D(A). (4.2.8)

The so called Yosida approximation Ah of A is defined by

Ahx :=x− Jhx

h∈ AJhx, x ∈ H, h > 0 (4.2.9)

where the inclusion follows by definition of Jh. It is not hard to show that Ah isLipschitz on H, for each h > 0.

There is a natural generalisation of the notion of monotone subsets A ⊂ H×H,namely the λ-monotone subsets, for λ ∈ R. A subset A ⊂ H ×H is λ-monotone,for λ ∈ R, if the following holds:

〈y1 − y2, x1 − x2〉 > λ|x1 − x2|2 ∀[x1, y1], [x2, y2] ∈ A. (4.2.10)

113


space P2(Rd)

If λ1 6 λ2, then any λ2-monotone subset A ⊂ H × H, is also λ1-monotone.Denoting I := (x, x)|x ∈ H to be the graph of the identity map on H, foreach λ ∈ R, A ⊂ H ×H is a l-monotone subset, if and only if the set A − λI :=[x, y−λx]| [x, y] ∈ A is monotone. Therefore, the most of the theory of maximalmonotone operators on Hilbert spaces presented in this section, is easily generalisedto maximal λ-monotone operators on Hilbert spaces. In this more general setting,one requires the resolvents to be defined for h > 0, such that 1 − λh > 0. Thistechnical relaxation of the definition (for λ < 0), is particularly interesting fortreatments of various PDE’s, and we will directly define λ-monotone operatorson the Wasserstein space P2(Rd), in Section 4.4. However, in order to explainthe main ideas, in this section we restrict our attention to maximal monotoneoperators, by giving an overview of the theory exposed in [17].

Remark 4.2.4. We also have the converse of the claim in (4.2.9). Namely forany x ∈ H and for any h > 0 the resolvent Jhx is the unique point z ∈ D(A) suchthat x−z

h ∈ Az. Indeed suppose that z ∈ D(A) satisfies this condition, then bymonotonicity of A we have

0 6 〈z − Jhx,x− zh− x− Jhx

h〉 = − 1

h|z − Jhx|2 (4.2.11)

thus z− Jhx = 0 holds. This is actually a proof of the fact that the resolvents arewell defined and single valued. We will use this observation to define resolvents inour context (see Section 4).

The following lemma is easy to prove and it is an essential component in theproof of the existence of the flow associated to A.

Lemma 4.2.5. For any h > 0 and for any x ∈ D(A) we have

|Aox| > |x− Jhxh

|. (4.2.12)

Proof. Since A is monotone and x−Jhxh ∈ AJhx, for any y ∈ Ax we have

1h〈x− Jhx, y −

x− Jhxh

〉 > 0,

so we can estimate

|y||x− Jhxh

| > 1h〈y, x− Jhx〉 > |x− Jhx

h|2.

The claim now follows by taking the infimum over all y ∈ Ax.

In the sequel, if a sequence (xn)n ⊂ H weakly converges to x ∈ H, then wewill denote xn

w→ x.

Proposition 4.2.6. Let H be a Hilbert space and let A ∈ H × H be a maximalmonotone operator. Suppose [xn, yn] ∈ A is a sequence such that xn

w→ x, ynw→ y

and lim supn〈xn, yn〉 6 〈x, y〉. Then [x, y] ∈ A and 〈xnyn〉 → 〈x, y〉.

114

Section 4.2

Proof. See [17] Proposition 2.5.

Remark 4.2.7. Notice that if xn → x ∈ H and ynw→ y ∈ H, then all the

conditions in the above proposition are satisfied and the strong-weak limit [x, y]of the sequence [xn, yn] must be in A too.

The construction of the contractive semigroup of solutions of the abstractCauchy problem associated to a maximal monotone operator A ⊂ H ×H definedin (4.1.1) can be found in Theoreme 3.1 of [17] and it is carried out as follows.One shows that for each u0 ∈ D(A) and h > 0 there is a unique solution of theequation

uh(t) = u0 +∫ t

0

Jh − Ih

uh(s)ds ∀t > 0. (4.2.13)

This equation is solved with the aid of the fact that Jh is a contraction for each h >0. Subsequently one shows that curves 0 6 t 7→ uh(t) convergence to the uniquesolution of (4.1.1). A careful reader of the proof in [17] can observe that much ofthe proof of this converge is argued with the aid of Lemma 4.2.5, Proposition 4.2.6and Remark 4.2.7.

Theorem 4.2.8. Let H be a Hilbert space and let A ⊂ H × H be a maximalmonotone operator. Then for each u0 ∈ D(A) the curve

u(0) := u0, u(t) := limn→+∞

(J tn

)nu0 for t > 0 (4.2.14)

has the following properties:

1. u(t) ∈ D(A) for each t > 0

2. 0 6 t 7→ u(t) is Lipschitz with Lipschitz constant |Aou0|

3. dudt +Au(t) 3 0 for a.e. t > 0

4. u(0) = u0

5. 0 6 t 7→ u(t) is differentiable from the right for each t > 0 and we have

du

dt++Aou(t) = 0 ∀t > 0 (4.2.15)

6. The function 0 6 t 7→ Ao(t) is continuous from the right and the function0 6 t 7→ |Aou(t)| is non-increasing.

7. If two functions 0 6 t 7→ u(t) and 0 6 t 7→ u(t) have properties 1,2, and 3above then we have the following estimate

|u(t)− u(t)| 6 |u(0)− u(0)| ∀t > 0 (4.2.16)

115


space P2(Rd)

Proof. The existence of curve with the listed properties is proven in [17] Theoreme3.1. The exponential formula (4.2.14) is very well known due to the celebratedmonograph of Crandall-Liggett [30].

The results about maximal monotone operators which we presented above willbe one guide in our investigations in following sections. An other guide are theresults of the next section.

4.3 Ambrosio-Gigli-Savare Frechet subdifferentialof geodesically convex functionals on P2(Rd)

In this section we recall the definition and several basic facts about the Frechet sub-differential introduced by L.Ambrosio, N.Gigli, and G.Savare of a regular geodesi-cally convex functionals on P2(Rd), which has been introduced in Chapter 10 of[5]. This definition is modeled on the properties of the Frechet subdifferential∂ϕ ⊂ H × H of a convex lower semi-continuous functional ϕ : H → (−∞,+∞]defined on a Hilbert space H:

ξ ∈ ∂ϕ(v)⇐⇒ v ∈ D(ϕ), lim infw→v

ϕ(w)− ϕ(v)− 〈ξ, w − v〉|w − v|

> 0 (4.3.1)

or equivalently

ξ ∈ ∂ϕ(v)⇐⇒ v ∈ D(ϕ), ϕ(w) > ϕ(v) + 〈ξ, w− v〉+ o(|w− v|) as w → v. (4.3.2)

Let us recall these properties first.

• A. Euler equation for quadratic perturbations: If vh is a minimizer of

w 7→ Φ(h, v, w) := ϕ(w) +1

2h|w − v|2, h > 0, v ∈ H,

thenvh ∈ D(∂ϕ), and − vh − v

h∈ ∂ϕ(v) (4.3.3)

If ϕ is moreover λ-convex for some λ ∈ R, then ∂ϕ also has the followingproperties.

• B. Characterization by variational inequalities and monotonicity:

ξ ∈ ∂ϕ(v)⇐⇒ ϕ(w) > ϕ(v) + 〈ξ, w − v〉+λ

2|w − v|2 ∀w ∈ H. (4.3.4)

In particular

ξj ∈ ∂ϕ(vj), j = 1, 2⇒ 〈ξ1 − ξ2, v1 − v2〉 > λ|v1 − v2|2, ∀v1, v2 ∈ D(∂ϕ)(4.3.5)

116

Section 4.3

• C. The strong weak closure: For any two sequences vn, ξn ∈ H we have

ξn ∈ ∂ϕ(vn), vn → v, ξn → ξ ⇒ ξ ∈ ∂ϕ(v), ϕ(vn)→ ϕ(v). (4.3.6)

Modeled on this property the authors of [5] call a functional ϕ regular if

ξn ∈ ∂ϕ(vn), φn := ϕ(vn)vn → v, ξn

w→ ξ, φn → φ

=⇒ ξ ∈ ∂ϕ(v), φ = ϕ(v).

• D. Minimal selection and slope: If ϕ is regular (in particular if it isλ−convex) then for each v ∈ D(ϕ) the metric slope

|∂ϕ|(v) := lim supw→v

(ϕ(v)− ϕ(w))+

|w − v|(4.3.7)

is finite if and only if ∂ϕ(v) 6= ∅ and if this holds then

|∂ϕ|(v) = min|ξ| : ξ ∈ ∂ϕ. (4.3.8)

• E. Chain rule: If v : (a, b)→ D(ϕ) is a continuous curve then

d

dtϕ(v(t)) = 〈ξ, v′(t)〉 ∀ξ ∈ ∂ϕ(v(t)), (4.3.9)

at each point t ∈ (a, b) where v and ϕv are differentiable and ∂ϕ(v(t)) 6= ∅.

In this and in the following sections for probability measures µ ∈ Pr2 (Rd), andν ∈ P2(Rd), the mapping rνµ : Rd → Rd denotes the unique optimal transport mapfrom µ to ν for the quadratic cost, and i denotes the identity map on Rd. We willonly consider functionals which satisfy the following condition

ϕ : P2(Rd)→ (−∞,+∞], ϕ 6≡ +∞ ϕ is lower semi-continuous and

D(|∂ϕ|) ⊂ Pr2 (Rd)(4.3.10)

The condition D(|∂ϕ|) ⊂ Pr2 (Rd) is needed for technical steps in our proofs. Itcan be eliminated at the cost of extra technicalities (see [5] Section 10.3). Wechoose to avoid such technicalities in this chapter (but see Section 4.9 for morecomments).

Before recalling the definition of the Frechet subdifferential given by L.Ambro-sio, N.Gigli and G.Savare in the context of P2(Rd), we devote the following threeparagraphs to recalling some basics about the differential calculus on manifolds,and in the setting of metric spaces. The discussion given in these lines is notrequired in order to follow the remainder of this chapter, and may be skipped atfirst reading.

If M is a smooth manifold then at each point p ∈ M there exists the tangentspace TpM and the derivative of a differentiable curve u : [a, b]→M at t ∈ [a, b] isper definition a vector u(t) ∈ Tu(t)M . Also, hand the derivative of a differentiable

117


space P2(Rd)

map Φ : M → R at a point p ∈M is per definition a linear map DpΦ : TpM → R,and if M has a Riemannian structure then by the Riesz representation theoremthere is a unique vector denoted ∇Φ(p) ∈ TpM that represents DpΦ, i.e. for eachx ∈ TpM we have DpΦ(x) = 〈∇Φ(p), x〉. Therefore the subdifferential ∂ϕ(p) ofany functional ϕ : M → (−∞,+∞] at p ∈M should be defined as a subset of TpMif M has Riemannian structure, and if ϕ is a lower semi-continuous, geodesicallyconvex functional, then for p ∈ D(ϕ), ξ ∈ TpM belongs to the subdifferential∂ϕ(p) of ϕ at p ∈M , if and only if

〈ξ, exp−1p q〉TpMϕ(p) 6 ϕ(q) ∀q ∈ D(ϕ). (4.3.11)

If X is a metric space then for any p ∈ X one can define the Alexandrov upperangle (see Chapter II) between each two geodesics emanating from p, which inducesa semi-distance on the space of directions at each p ∈ X (see Section 2 of ChapterII for details). The set of equivalence classes induced by this semi-distance definesa metric space, and its completion is called the space of directions and denoted bySp(X). If the first variation formula holds in X, then the Euclidean cone C0Sp(X)over Sp(X), reflects the local geometry of X (in the neighborhood of p), and it canbe used to recover many basic calculus formulas which we have in the Riemanniansetting. If X is a Riemannian manifold, then the Alexandrov upper angle is theusual Euclidean angle between geodesics, Sp(X) is just the unit sphere in TpXand C0Sp(X) coincides with TpX for each p ∈ X. Particular classes of spaceswhere the first variation formula holds are spaces with one sided curvature boundin the sense of Alexandrov, and in particular it holds on P2(Rd) (recall that theWasserstein-2 space is positively curved (see [5] Theorem 7.3.2). Moreover, if X isan Alexandrov space (and even in some other cases), one can give a useful analogueof the subdifferential of lower semi-continuous, geodesically convex functionals.

The space P2(Rd) is in many ways similar to a manifold. By [5] Theorem7.2.2, each geodesic in P2(Rd) is induced by an optimal transport plan, which is ameasure in P2(Rd ×Rd), and γε := (π1, π1 + επ2)#γ is an optimal transport planfor some ε > 0, i.e. that its support is cyclically monotone (see [42]). Howeveras we already mentioned we will restrict our considerations to the regular case,i.e. we assume (4.3.10). This implies that we only need the definition of ∂ϕ(µ)at regular measures µ, and since such a µ has the property that for any othermeasure ν ∈ P2(Rd) there is a unique optimal transport map from µ to ν, eachpoint in the abstract Euclidean tangent cone at µ as described above, is naturallyidentified with some element of the regular tangent cone TanµP2(Rd) (see also [5]Chapter 12 for a more detailed discussion about these matters).

Definition 4.3.1. (Frechet subdifferential on P2(Rd) of Ambrosio-Gigli-Savare)Let ϕ satisfy (4.3.10), and let µ ∈ D(|∂ϕ|). We say that ξ ∈ L2(µ; Rd) belongs tothe Frechet subdifferential ∂ϕ(µ) of ϕ at µ if

lim infν→µ

ϕ(ν)− ϕ(µ)−∫

Rd〈ξ, rνµ − i〉dµ

W2(µ, ν)(4.3.12)

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Section 4.3

or equivalently

ϕ(ν)− ϕ(µ) >∫〈ξ, rνµ − i〉dµ+ o(W2(µ, ν)). (4.3.13)

When ξ ∈ ∂ϕ(µ) also satisfies

ϕ(r#µ)− ϕ(µ) >∫〈ξ, r − i〉dµ+ o(‖r − i‖L2(µ;Rd))) (4.3.14)

for any r ∈ L2(µ; Rd) we say that ξ is a strong subdifferential and in that case wedenote ξ ∈ ∂stϕ(µ). We will abbreviate the terminology Frechet subdifferential tosubdifferential at times. By D(∂ϕ) we denote the set of µ such that ∂ϕ(µ) 6= ∅and the set of pairs [µ, ξ] such that µ ∈ D(∂ϕ) and ξ ∈ ∂ϕ(µ) is denoted by ∂ϕ.D∂stϕ and ∂stϕ are defined in an analogous way.

Remark 4.3.2. Definition 4.3.1 is inspired by the definition of Frechet subdiffer-ential as these authors explained in their work. However Frechet subdifferentialis an object defined in the context of convex analysis on Hilbert spaces. Whilethe definition bellow is a part of the theory of optimal transportation and gra-dient flows, and contains various steps, which are highly novel with respect tothe classical theory on Hilbert spaces. In particular, this definition is foundedon the identification of the tangent spaces in P2(Rd) with gradients of C∞c (Rd),and with the continuity equation in P2(Rd) (see [5] Theorem 8.3.1). Therefore itseems more appropriate, at least to the author of this thesis, to call these subdif-ferentials ’Ambrosio-Gigli-Savare subdifferentials’ or more shortly ’AGS subdiffer-entials’, than the rather long and not quite accurate term ’Frechet subdifferentialsof the functionals defined on the Wasserstein space’. Henceforth, we adopt theterm ’AGS subdifferentials’, throughout the remainder of this thesis.

Notice that the AGS subdifferential of a regular functional ϕ : P2(Rd) →(−∞,+∞] at µ ∈ P2(Rd), may equivalently be required to be a member ofTanµP2(Rd), such that (4.3.12) holds, since any ξ ∈ L2(µ; Rd) is a sum of twomutually orthogonal elements, one in TanµP2(Rd) and the other in its orthogonalcomplement, and by [5] Theorem 8.5.1 vectors of the form rνµ− i are tangent. It isclear that if ξ ∈ L2(µ; Rd) is a strong subdifferential then also ξ ∈ ∂ϕ(µ). In thesequel the subset of strong subdifferentials of ϕ at µ will be denoted by ∂stϕ(µ). Itshould be obvious that the above AGS subdifferential resembles its Hilbert spacecounterpart with great analogy. In the sequel we will assume that for some h∗ > 0

ν 7→ Φ(h, µ, ν) :=1

2hW 2

2 (µ, ν) + ϕ(ν) admits at least

one minimum point µh, ∀h ∈ (0, h∗), ∀µ ∈ P2(Rd)(4.3.15)

and in some lemmas below also that for some λ ∈ R

ϕ is λ-convex on geodesics. (4.3.16)

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space P2(Rd)

Recall that a functional ϕ : P2(Rd) → (−∞,+∞] is λ-geodesically convex, if foreach µ0, µ1 ∈ D(ϕ), there is a geodesic [0, 1] 3 t 7→ µt that joins µ0 with µ1, suchthat

ϕ(µt) 6 (1− t)ϕ(µ0) + tϕ(µ1)τ(1− t)W 22 (µ0, µ1), ∀t ∈ [0, 1]. (4.3.17)

The above definition of Frechet subdifferential exhibits great analogy with itsHilbert space counterpart. Moreover in [5] authors show that it possesses proper-ties as in A, B, C, D, and E above.

Before proceeding, we need to recall Definition 5.4.3 in [5] of weak convergenceof vectors belonging to different Lp.

Definition 4.3.3. Let e1, ..ed ∈ Rd denote the standard orthonormal basis for Rd,let (µn)n ⊂ P(Rd) be weakly converging to µ ∈ P(Rd), and let vn ∈ L1(µn; Rd),for n ∈ N. We say that vn weakly converge to v ∈ L1(µ; Rd), if

limn→+∞

∫Rdψ(x)〈ej , vn(x)〉dµn(x) =

∫Rdψ(x)〈ej , v(x)〉dµ(x), (4.3.18)

for each ψ ∈ C∞c (Rd), and j = 1, ...d. We say that (vn)n strongly converges tov ∈ Lp(µ; Rd), p > 1, if (4.3.18) holds, and

lim supn→+∞

|vn|Lp(µn;Rd) 6 |v|Lp(µ;Rd). (4.3.19)

In this chapter, we will only consider the case where p = 2 in Definition 4.3.3,and since there is no ambiguity we will omit p in the notation. The basic propertiesof the weak convergence given in Definition 4.3.3, are given in [5] Theorem 5.4.4.

A. Euler equation for quadratic perturbations in P2(Rd)

Lemma 4.3.4. Let ϕ satisfy (4.3.10) and (4.3.15). Moreover for h∗ > h > 0 andµ ∈ P2(Rd) let µh be a minimizer of ν 7→ Φ(h, µ, ν). Then µh ∈ D(|∂ϕ|) and

rµµh − ih

∈ ∂stϕ(µh). (4.3.20)

Proof. See [5] Lemma 10.1.2.

The analogy with the property A. in the Hilbert space setting and more gen-erally with (4.2.9) is that

rµµh−ih should be interpreted as ’µ−µhh ’ while µh = Jhµ.

However, in Lemma 4.3.4 only one direction of the claim in the Hilbert space set-ting is proven (see Remark 4.2.4. Proving that for any µ and h > 0, Jhµ is the

unique point ν ∈ P2(Rd) that has the property thatrνµh−ih ∈ ∂ϕ(µh), will give a an

alternative characterization of the resolvents in terms of the subdifferential. Sincethis characterisation is in terms of ∂ϕ, which is an operator (see Section 4.4), it issuitable for generalization of the existing theory of gradients flows, to the theoryto maximal monotone operators, which we will develop in the following sections..We will prove this converse under the assumption that the considered functionalis convex along generalized geodesics in the next section.

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Section 4.3

B. Characterization by variational inequalities and monotonicity inP2(Rd)

Just as in the Hilbert space case subdifferentials of λ-convex functionals are λ-monotone in an appropriate sense. With the aid of this property one can showthat the induceed semigroup is semi-contractive.

Lemma 4.3.5. Suppose that ϕ satisfies (4.3.10) and (4.3.16). Then a vectorξ ∈ L2(µ; Rd) belongs to ∂ϕ(µ) if and only if

ϕ(ν)− ϕ(µ) >∫

Rd〈ξ, rνµ − i〉dµ+

λ

2W 2

2 (µ, ν), ∀ν ∈ D(ϕ). (4.3.21)

In particular if µj ∈ D(|∂ϕ|), and ξj ∈ ∂ϕ(µj) for j = 1, 2 , then∫Rd〈ξ rµ2

µ1− ξ2, rµ2

µ1− i〉dµ1 > λW 2

2 (µ1, µ2). (4.3.22)

Proof. See [5] Section 10.1.1 B.

C. Convexity and strong-weak closure in P2(Rd)

At the end of the previous section we mentioned that one of the key building blocksof the classical proof of existence of the solutions of the Cauchy problem (4.1.1)(also in the special case where A = ∂ϕ) is the fact that A is strong-weak closed inH × H, see Proposition 4.2.6 and Remark 4.2.7. It is not surprising that it alsoplays an essential role in the proof of existence of the gradient flow paths in thesetting of P2(Rd). Contrary to the Hilbert space case, in P2(Rd) tangent spacesat two distinct points are not isometrically isomorphic in a canonical way, and anotion of convergence of sequences of vectors belonging to different L2 spaces isrequired. We will use the notion of weak convergence of a sequence of vectors thatbelong to different L2 spaces, given in Definition 4.3.3.

Lemma 4.3.6. Let ϕ satisfy (4.3.10) and (4.3.16), let µn be converging to µ ∈D(ϕ) in P2(Rd) and let ξn ∈ ∂ϕ(µn) satisfy

supn

∫Rd|ξn|2 dµn < +∞ (4.3.23)

and let ξn converge to ξ ∈ L2(µ; Rd) weakly according to Definition 4.3.3. Thenξ ∈ ∂ϕ(µ).


Properties A, B and C above are of crucial importance for the investigationsof this chapter. Properties D and E are not relevant at all since if one considersan operator, in general there is no object whose properties can be related to theproperties D and E.

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space P2(Rd)

D. Minimal selection and the metric slope in P2(Rd)

Lemma 4.3.7. Let ϕ satisfy (4.3.10), (4.3.15) and (4.3.16). Then for any µ ∈P2(Rd) we have µ ∈ D(|∂ϕ|) if and only if ∂ϕ(µ) 6= ∅ and

|∂ϕ|(µ) = min|ξ|L2(µ;Rd) : ξ ∈ ∂ϕ(µ). (4.3.24)

Since ∂ϕ(µ) is a closed convex subset of L2(µ; Rd) there exists a unique vectorξ ∈ ∂ϕ(µ) at which the minimum in (4.3.24) is attained, and we will denote thisvector by ∂ϕo(µ).


E. Chain rule in P2(Rd)

Lemma 4.3.8. Let ϕ satisfy (4.3.10), (4.3.15) and (4.3.16), and let µ : (a, b)→µt ∈ D(ϕ) ⊂ P2(Rd) be an absolutely continuous curve with the tangent velocityvector field (vt)t∈(a,b). Let Λ ⊂ (a, b) be the set of points t ∈ (a, b) such that

(i) |∂ϕ|(µt) < +∞;(ii) ϕ µ is approximately differentiable at t;

(iii) condition (8.4.6) of Proposition 8.4.6 in [5] holds.

Then we have

ddtϕ(µt) =

∫Rd〈ξt, vt〉dµt ∀ ξ ∈ ∂ϕ(µt), ∀t ∈ ∂ϕ(µt). (4.3.25)

Moreover if ∫ b

a

|∂ϕ|(µt)|µ|(t) dt < +∞ (4.3.26)

then the map t 7→ ϕ(µt) is absolutely continuous and (a, b) \ Λ is L1 negligible.

Proof. See [5] Section 10.1.2 E.

Let us recall that a functional ϕ : P2(Rd) → (−∞,+∞] is λ-convex alonggeneralized geodesics for λ ∈ R if for any µ0, µ1 ∈ D(ϕ) and for any σ ∈ P2(Rd)there is a 3-plan as in (4.1.5) such that the curve [0, 1] 3 t 7→ γ(t) defined in (4.1.6)satisfies

ϕ(γ(t)) 6 (1− t)ϕ(µ0) + ϕ(µ1)− λ

2W 2

2 (µ0, µ1) ∀ t ∈ [0, 1]. (4.3.27)

If a proper lower semi-continuous functional satisfies this convexity property (whichis stronger than (4.3.16) by [5] Remark 9.2.8) together with a non degeneracycondition (i.e it must be bounded from below on some ball in P2(Rd)) then itsresolvents (4.3.15) are defined everywhere and the discrete Euler scheme converges

122

Section 4.4

to the unique solution [0,+∞) 3 t 7→ µt ∈ D(|∂ϕ|) of the EVI starting at eachµ ∈ D(ϕ), i.e.

ddt

12W 2

2 (µt, σ) +λ

2W 2

2 (µt, σ) + ϕ(µt) 6 ϕ(σ) ∀σ ∈ D(ϕ). (4.3.28)

Moreover, as we will see in the next section, the subdifferential of such a func-tional satisfies the appropriate analogue of Remark 4.2.4. We will be consideringfunctionals that satisfy the following properties:

ϕ :P2(Rd)→ (−∞,+∞], D(|∂ϕ|) ⊂ Pr2 (Rd), ϕ 6≡ +∞ϕ is lower semi-continuous andϕ is λ-convex along generalized geodesics for some λ ∈ R∃r∗ > 0, ∃ν ∈ P2(Rd) such that inf

W2(µ,ν)6r∗ϕ(µ) > −∞.

(4.3.29)

4.4 Maximal λ-Monotone Operators in general-ized sense, associated Cauchy problems andthe Resolvents on (P2(R2),W2)

In this section we will give our definition of regular λ-monotone and maximal λ-monotone operators in generalized sense on P2(Rd), which we call λ-MGR andλ-MMGR operators, respectively, where λ is a real number. Moreover we will in-troduce the associated abstract Cauchy problem and also define the correspondingresolvents. Let us start by introducing some notation. Throughout the remainderof this chapter we denote

L2(Rd) :=⊔

µ∈P2(Rd)

L2(µ; Rd) (4.4.1)

which we consider to be a disjoint union of sets. Furthermore for any subsetA ⊂ P2(Rd)× L(Rd) we denote the domain of A by

D(A) := µ ∈ P2(Rd)| ∃ξ ∈ L2(Rd) such that [µ, ξ] ∈ A, (4.4.2)

and its image by

Im(A) = ξ ∈ L2(Rd)| ∃µ ∈ P2(Rd) such that [µ, ξ] ∈ A. (4.4.3)

Moreover for µ ∈ D(A) we denote

A(µ) := ξ ∈ L2(Rd)| [µ, ξ] ∈ A (4.4.4)

A subset ∅ 6= A ⊂ P2(Rd)× L2(Rd) is called a Wasserstein−2 operator if for each[µ, ξ] ∈ A we have that ξ ∈ L2(µ,Rd), and it is called a regular Wasserstein-2 operator if D(A) ⊂ Pr2 (Rd), or shortly regular operators. The elements of a

123


space P2(Rd)

regular operator A will typically be denoted by [µ, ξ], [ν, η] etc., where µ, ν denotea measures in P2(Rd), and ξ ∈ L2(µ; Rd), and η ∈ L2(ν,Rd) denote vectors. Allthe operators we consider in the sequel are Wasserstein-2 regular operators unlesswe specify otherwise.

If H is a Hilbert space and ∅ 6= A ⊂ H × H is an operator (not necessarilymonotone), one can associate the evolution problem (4.1.1) with it, whose solutionsare curves which are differentiable a.e. on [0,+∞). A natural analogue of theCauchy problem in the present setting is given in the following definition.

Definition 4.4.1. Let A ⊂ P2(Rd)×L2(Rd) be a regular operator. A continuouscurve 0 6 t 7→ µ(t) with µ(0) := µ0 ∈ D(A) is a solution of the abstract Cauchyproblem associated to A with initial value µ0 = µ0 if there is a Borel vector field[0,+∞)× Rd 3 (t, x) 7→ vt(x) ∈ Rd such that∫ T

0

|vt|2L2(µt;Rd) dt < +∞, ∀T > 0, (4.4.5)∫ ∞0

∫Rd∂tψ(t, x) + 〈∇xψ(t, x), vt〉dµt(x) dt+

∫Rdψ(0, x) dµ0(x) = 0,

∀ψ ∈ C∞c ([0,+∞)× Rd),(4.4.6)

and such that[µt,−vt] ∈ A for L1 a.e. t > 0 (4.4.7)

Notice that by [5] Theorem 8.3.1, given a regular operator A, the solutionsof the associated abstract Cauchy problem posed in Definition 4.4.1, are of classAC2((0, T ); (P2(Rd),W2)). Notice also that, as in [5] Theorem 8.3.1, the vectorfield (t, x) 7→ v(x) only needs to be defined for L1-a.e. t ∈ [0, T ].

Throughout the remainder of this chapter the dimension d ∈ N of the Euclideanspace is arbitrary but fixed and we will use the following notation: for r ∈ N andx ∈ Rdr abusing the notation slightly we write

x = (x0, x1, ..., xr−1), x0, x1, ..., xr−1 ∈ Rd. (4.4.8)

Moreover for N 3 k 6 r and for an increasing sequence j1, j2, ..., jk ∈ 0, 1, .., r−1of distinct integers we denote

πj1,j2,...,jkr : Rdr → Rdk, πj1,j2,...,jkr (x0, x1, ..., xr−1) := (xj1 , xj2 , ..., xjk). (4.4.9)

In particular, we will consider the case k = 2, r = 3 and in this case we will omitthe subscript 3 in (4.4.9) for each of the possible choices k = 1, 2, 3. The followingdefinition and also the remark below the definition are essentially borrowed from[5] (see Definition 9.2.2 and Remark 9.2.3 there).

Definition 4.4.2. Let σ, µ1, µ2 ∈ P2(Rd). A 3plan γ ∈ P2(Rd × Rd × Rd) is saidto connect µ1 with µ2 through σ if

π0,1# γ ∈ Γo(σ, µ1), π0,2

# γ ∈ Γo(σ, µ2). (4.4.10)

As we explained above the subscript 3 is omitted. In such a situation we will referto σ as the base point, or base measure or reference measure of the triple σ, µ1, µ2.

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Section 4.4

In the sequel, we will use the following lemma extensively.

Lemma 4.4.3. For any triple of measures σ, µ1, µ2 ∈ P2(Rd) if the referencemeasure is regular, i.e. if σ ∈ Pr2 (Rd), then the 3-plan γ := (i × rµ1

σ × rµ2σ )#σ is

the unique 3-plan γ ∈ P2(R3d) which connects µ1 with µ2 through σ.

Proof. Follows by [5] Lemma 5.3.2.

We will borrow the following notation from [5]. For measures µ1, µ2 ∈ P2(Rd)and a 3-plan γ ∈ P2(R3d) such that πj#γ = µj for j = 1, 2 we denote:

W 2γ :=

∫R3d|x1 − x2|2 dγ(x0, x1, x2) (4.4.11)

Observe that in such a situation π1,2# γ ∈ Γ(µ1, µ2) holds so that we have

W2(µ1, µ2) 6 Wγ(µ1, µ2). (4.4.12)

The following lemma is Lemma 5.2.4 in [5] and we will need it in the sequel.

Lemma 4.4.4. Let (νn)n ⊂ P2(R2d) be a sequence weakly converging to ν ∈P(R2d) with

supn

∫R2d

(|x1|2 + |x2|2) dνn(x1, x2) < +∞. (4.4.13)

If either (π12)#νn or (π2

2)#νn has uniformly integrable 2-moments, then

limn→+∞

∫R2d〈x1, x2〉dνn =

∫R2d〈x1, x2〉dν. (4.4.14)

As we already mentioned in the previous section (see property B of the AGSsubdifferential) two general methods to construct gradient flows on P2(Rd) areproposed in [5]. One method applies under a compactness assumption of the con-sidered functional which guarantees existence of solutions of one step backwardEuler approximation problem as well as convergence of the obtained discrete so-lutions to the solution of the EVI. The other method applies if the consideredfunctional is convex along generalized geodesics and it also ensures existence ofresolvents and the convergence of approximations to the solution. We will gen-eralize the second approach. This approach exploits that (−1)-convexity of W 2

2

along generalized geodesics (see (4.1.5) and (4.1.6) for definition).Observe that the natural definition of λ-monotonicity of a multivalued vector

field A ⊂M × TM (thus ∀p ∈ D(A) Ap ⊂ TpM holds by definition) where M is aHadamard manifold, i.e. a simply connected Riemannian manifold with globallynon-positive sectional curvature) is the following

〈ξ1, exp−1p1

(p2)〉Tp1M+ 〈ξ2, exp−1

p2(Tp1M)〉Tp2

+ λd2(p1, p2) > 0

∀ [p1, ξ1], [p2, ξ2] ∈ A.(4.4.15)

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space P2(Rd)

Here we used the usual notation for the inverse of the exponential map, i.e. fortwo points p1, p2 ∈M , exp−1

p1(p2) ∈ Tp1M denotes the derivative at 0 of the unique

geodesic c : [0, 1]→M such that c(0) = p1 and c(1) = p2. The continuity equation[5] Theorem 8.3.1 characterizes the Euclidean tangent cones of P2(Rd) (which isan Alexandrov space), therefore the following definition seems appropriate.

Definition 4.4.5. Let λ ∈ R. A regular operator A is called λ-monotone if foreach pair of its elements [µ1, ξ1], [µ2, ξ2] ∈ A we have∫ ⟨

ξ1 − ξ2 rµ2µ1, rµ2µ1− i⟩

dµ1 + λW 22 (µ1, µ2) 6 0. (4.4.16)

A regular operator A is called λ-monotone in generalized sense and for each[µ1, ξ1], [µ2, ξ2] ∈ A and for each σ1, σ2 ∈ P2(Rd) there are 3-plans γ1 ∈ P2(R3d)and γ2 ∈ P2(R3d) that connect µ1 and µ2 through σ1 and σ2, respectively, andsuch that we have∫

〈ξ1(x1), x2 − x1〉dγ1 +∫〈ξ2(x2), x1 − x2〉dγ2

+λ

2W 2γ1

(µ1, µ2) +λ

2W 2γ2

(µ1, µ2) 6 0(4.4.17)

where we implicitly used notation (4.4.8). Regular operators which are λ-monotonein generalized sense will shortly be denoted by the acronym λ-MGR.

Remark 4.4.6. Notice that any λ-MGR operator A is also a λ-monotone oper-ator. Indeed, for any [µ1, ξ1], [µ2, ξ2] ∈ A, choosing the base measures σj := µj ,for j = 1, 2 in Definition 4.4.5, we must have that x0 = x1 γ1-a.e., and x0 = x2

γ2-a.e., hence π1,2# γj ∈ Γo(µ1, µ2) for j = 1, 2, and Wγj = W2 for j = 1, 2, and the

conclusion follows. Notice moreover that for λ1, λ2 ∈ R such that λ1 6 λ2, anyregular λ2-monotone operator A is also a regular λ1-monotone operator. Likewise,any λ2-MGR operator is also a λ1-MGR operator.

Next we show that we may restrict our scope to base measures σ1, σ2 in Defi-nition 4.4.5 to regular ones, without reducing the generality.

Proposition 4.4.7. Let λ ∈ R and let A be a regular operator. Suppose thatfor each [µ1, ξ1], [µ2, ξ2] ∈ A and σ1, σ2 ∈ Pr2 (Rd) there are 3-plans γ1 and γ2

in P2(R3d) which connect µ1 with µ2 through σ1 and σ2, respectively, such that(4.4.17) holds. Then A is a λ-MGR operator. Moreover if A is a λ-MGR operatorthen for any [µ1, ξ1], [µ2, ξ2] ∈ A and for any σ ∈ P2(Rd) there is a 3-plan γ whichconnects µ1 with µ2 through σ such that∫

R3d〈ξ1(x1)− ξ2(x2), x2 − x1〉dγ + λW 2

γ (µ1, µ2) 6 0. (4.4.18)

Proof. Let us argue the first claim first. Pick [µ1, ξ1], [µ2, ξ2] ∈ A and σ1, σ2 ∈P2(Rd) and consider two sequences Pr2 (Rd) 3 σnj → σj for j = 1, 2. By the

126

Section 4.4

Lemma 4.4.3 we have that for j = 1, 2 and for each n ∈ N the 3-plan γnj :=(i× rµ1

σnj× rµ2

σnj)#σ

nj is the only plan which connects µ1 with µ2 through σnj . Define

the 3-plans γnj := (ξj πj , π1, π2)#γnj for j = 1, 2 and n ∈ N. Well now by

assumption we have

0 >∫

R3d〈ξ1(x1), x2 − x1〉dγn1 +

∫R3d〈ξ2(x2), x1 − x2〉dγn2

+λ

2W 2γn1

(µ1, µ2) +λ

2W 2γn1

(µ1, µ2)

=∫

R3d〈x0, x2 − x1〉dγn1 +

∫R3d〈x0, x1 − x2〉dγn2

+λ

2W 2γn1

(µ1, µ2) +λ

2W 2γn1

(µ1, µ2)

(4.4.19)

for each n ∈ N. Since convergence of a sequence in P2(Rd) implies convergence ofthe corresponding second moments (see [5] Remark 7.1.11), we have for j = 1, 2∫

R3d|x|2 dγnj (x) =

∫Rd|x0|2 dσnj (x0) +

∫Rd|x1|2 dµ1(x1) +

∫Rd|x2|2 dµ2(x2)

n→+∞−→∫

Rd|x0|2 dσj(x0) +

∫Rd|x1|2 dµ1(x1) +

∫Rd|x2|2 dµ2(x2).

(4.4.20)Hence both sequences (γnj )n, j = 1, 2, have bounded second moments (in n) andtherefore they are relatively weakly compact. The same conclusion holds for thesequences (γnj )n, j = 1, 2, since their second moments are constant:

N 3 n 7→∫

R3d|x|2 dγnj (x)

=∫

Rd|ξj |2(xj) dµj(xj) +

∫Rd|x1|2 dµ1(x1) +

∫Rd|x2|2 dµ2(x2).

(4.4.21)

Hence there is a subsequence (nk)k of natural numbers such that each of the foursequences (γnkj )k and (γnkj )k, j = 1, 2, weakly converge to some measures denotedγj and γj , j = 1, 2. We will abuse notation slightly and omit the subscript k.Clearly

π0#γj = σj , π1

#γj = µ1, π2#γj = µ2,

π1#γj = µ1, π2

#γj = µ2

(4.4.22)

holds for j = 1, 2. Moreover, since the weak convergence of a sequence of measuresin P(R3d) implies weak convergence of each of the corresponding marginals, wehave

(ξj)#µj = (ξj πj)#γnj = π0

#γnj → π0

#γj n→ +∞ (4.4.23)

weakly in P(Rd), hence

x0 = ξj(xj) γj -a.e. for j = 1,2 (4.4.24)

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space P2(Rd)

Now (4.4.22), (4.4.23) and (4.4.24) imply that∫R3d〈x0, x2 − x1〉dγ1 =

∫R3d〈ξ1(x1), x2 − x2〉dγ1 (4.4.25)

and ∫R3d〈x0, x2 − x1〉dγ2 =

∫R3d〈ξ2(x2), x1 − x2〉dγ2. (4.4.26)

On the other hand we have that∫R3d〈ξ1(x1), x2 − x1〉dγn1 +

∫R3d〈ξ2(x2), x1 − x2〉dγn2

=∫

R3d〈x0, x2 − x1〉dγn1 +

∫R3d〈x0, x1 − x2〉dγn2

−→∫

R3d〈x0, x2 − x1〉dγ1 +

∫R3d〈x0, x1 − x2〉dγ2 as n→ +∞,

(4.4.27)

where the convergence follows with the aid of Lemma 4.4.4, and that lemma alsoimplies that

W 2γnj

(µ1, µ2) =∫

R3d|x1 − x2|2 dγnj

=∫

Rd|x1|2 dµ1(x1) +

∫Rd|x2|2 dµ2(x2)− 2

∫R2d〈x1, x2〉dπ1,2

# γnj

−→∫

Rd|x1|2 dµ1(x1) +

∫Rd|x2|2 dµ2(x2)− 2

∫R2d〈x1, x2〉dπ1,2

# γj

=W 2γj (µ1, µ2)

(4.4.28)

for j = 1, 2. In light of (4.4.19) we have now proved the first claim.The second claim, i.e. (4.4.18) follows by approximatingγj by regular measures,

applying Lemma 4.4.3, and completing the proof by similar arguments.

Next we need to give a definition of the resolvents associated to a λ-MGRoperator A, and these resolvents should at least be defined for each µ ∈ D(A) andfor all sufficiently small h > 0. These resolvents will play the role of the Eulerbackward approximations in solving the evolution problem posed in Definition4.4.1.

The property A. of the subdifferential of regular λ-convex functionals definedon P2(Rd) (see Section 4.3) suggests that it is natural to require that for anyµ ∈ P2(Rd) and h > 0 the (to be defined) resolvent Jhµ associated to A satisfies

rµµh − ih

∈ A(µh) where µh = Jhµ. (4.4.29)

We show that for any h > 0 and µ ∈ P2(Rd), there can be at most one pointµh ∈ D(A) which satisfies (4.4.29).

128

Section 4.4

Lemma 4.4.8. Let λ ∈ R and let A be a λ-MGR operator. Let moreover h > 0be such that 1 + 2hλ > 0 and fix ν ∈ P2(Rd). Then [µ1,

rνµ1−ih ], [µ2,

rνµ2−ih ] ∈ A

implies µ1 = µ2.

Proof. By Proposition 4.4.7 there exists a 3-plan γ which connects µ1 with µ2

through ν, such that (4.4.18) holds with σ := ν and ξj :=rνµj−ih for j = 1, 2 there.

Moreover since D(A) ⊂ Pr2 (Rd) by definition, the uniqueness of optimal transportplans for regular measures (see [42]) and (4.4.10) imply that π0,j

# γ = (rνµj × i)#µjfor j = 1, 2. Then by assumption we have x0 = xj + hξj(xj) γ -a.e. for j = 1, 2,and we estimate

0 =∫|x0 − x0|2dγ =

∫|x1 + hξ1(x1)− x2 − hξ2(x2)|2dγ

=∫|x1 − x2|2dγ + 2h

∫〈ξ1(x1)− ξ2(x2), x1 − x2〉dγ

+ h2

∫|ξ1(x1)− ξ2(x2)|2dγ >

>∫|x1 − x2|2dγ + 2hλW 2

γ (µ1, µ2)

= (1 + 2hλ)W 2γ (µ1, µ2) > (1 + 2hλ)W 2

2 (µ1, µ2),(4.4.30)

and since 1 + 2hλ > 0 is assumed the conclusion of the lemma follows.

Now we are in position to define the resolvents associated to A.

Definition 4.4.9. Let λ ∈ R and let A be a λ-MGR operator. For h > 0 andµ ∈ P2(Rd) if µh ∈ D(A) satisfies

[µh,rµµh − ih

] ∈ A (4.4.31)

then the point µh is called the h resolvent of µ associated to A and we will denote

Jhµ := µh (4.4.32)

By Lemma 4.4.8 there can be at most one such point µh hence this definition isunambiguous.

Finally we give our definition of regular maximal λ-monotone operators ingeneralized sense. Recall Definition 4.3.3.

Definition 4.4.10. A λ-MGR operator A is said to be maximal if the followingtwo conditions hold:

1. There is an h0 > 0 such that for each h < h0 and for each ν ∈ D(A) there isa pair [µ, ξ] ∈ A such that

ξ =rνµ − ih

. (4.4.33)

129


space P2(Rd)

2. If [µn, ξn] ∈ A for n ∈ N and for some µ ∈ P2(Rd) and ξ ∈ L2(µ,Rd) onehas µn → µ in P2(Rd) and ξn → ξ weakly, while supn ‖ξn‖L2(µn,Rd) < +∞,then [µ, ξ] ∈ A.

A is said to be complete if for each h < h0 and for each ν ∈ P2(Rd) there is apair [µ, ξ] ∈ A such that (4.4.33) holds. Maximal λ-MGR operators will shortlybe denoted by λ-MMGR and complete λ-MMGR will be denoted by λc-MMGR.

Let us give some comments about Definition 4.4.10. In the Hilbert space settinga maximal monotone operator is by definition a maximal element with respectto set inclusion in the class of monotone operators on the Hilbert space underconsideration. Subsequently one proves that this assumption is equivalent withthe assumption that resolvents are well defined on the whole space and for eachh > 0. Such equivalence might be rather hard to prove in our setting for severalgeometric reasons.3

Firstly there is no natural isometric isomorphism between tangent spaces atdifferent points in P2(Rd). Our situation is even more delicate since we are work-ing with monotonicity in generalized sense, which is a different formulation thangeodesic monotonicity. A fixed point type of argument as used in [65] could notbe applied directly, since P2(Rd) is not a CAT(0) space.

Secondly, tangent spaces TanP2(Rd) are metric completions of the sets

t(r − i) : t > 0, r : Rd → Rd, (i, r)#µ ∈ Γo(µ, r#µ), (4.4.34)

which gives a structural disadvantage with respect to Hilbert spaces, or even withrespect to Riemannian manifolds. Namely, for each point x in a Hilbert space H,each element of the tangent space z ∈ TxH ∼= H, is a derivative of the geodesic[0, 1] 3 t 7→ x+ tz, which emanates from x. In P2(Rd) however, a vector ξ in thetangent space TanµP2(Rd) at µ ∈ P2(Rd), may be a derivative of a curve, whilethe curve [0, ε] 3 t 7→ (i + tξ)#µ is not a geodesic for any ε > 0. In spite of thefact that the Wasserstein derivatives along absolutely continuous curves in P2(Rd)are only L1-a.e. defined, this geometric pathology may give difficulties in provingan analogue of Proposition 4.2.1.

Thirdly, geodesics in P2(Rd) can not be extended to a longer geodesic, ingeneral. This gives the following difference with respect to the Hilbert spacesetting. Suppose A ⊂ H×H is a monotone subset of a Hilbert space H, x, z ∈ H,h > 0 and suppose that [z, x−zh ] ∈ A (i.e. ξ := x−z

h ∈ Az). Then for any h1 > 0the point xh1 := z + h1ξ = z + h1

h (x − z) is a point on the line (i.e. the affinespace of dimension 1) which is determined by x and z. Moreover Jh1xh1 = z orsaid differently ξ = xh1−z

h1∈ Az for each h1 > 0. On the other hand suppose that

ν ∈ P2(Rd) and that for some h > 0 and µ ∈ D(A) we have that [µ, rνµ−ih ] ∈ A, so

that Jhν = µ. On the other hand, a geodesic between points µ, ν ∈ P2(Rd), cannot be extended (to a geodesic ray, or line), in general, since the map ξ := rνµ−i

h

3As a matter a fact in [65] the authors analyze monotone operators on Hadamard manifoldsand prove equivalence as in Proposition 4.2.1 only for operators whose domain is the whole space.

130

Section 4.5

may not have the property that for each h1 > h the map i + h1ξ is an optimaltransport map, mapping µ to (i+ h1ξ)#µ. Roughly speaking, this means that inorder to compensate, we need to add more pairs into A. However if [µ, ξ] ∈ ∂sϕ issuch that for each h > 0, (i, i + hξ)#µ ∈ Γo(µ, (i + hξ)#µ) holds, i.e if the curve0 6 h 7→ (i+ hξ)#µ is a geodesic line in P2(Rd), then by McCann’s theorem, foreach h > 0 the mapping i, i+ hξ, hence also 1

h (i+ hξ) = ih + ξ, is µ a.s. cyclically

monotone. Thus for each h > 0, and x1, ..., xn ∈ suppµ (denoting xn+1 := x1, andsuppµ to be the support of the positive measure µ) we have that

n∑j=1

⟨(i

h+ ξ′

)(xj).xj+1 − xj

⟩6 0, (4.4.35)

Taking the limit h → +∞ in (4.4.35), we obtain that ξ′ is cyclically monotone,hence (i, ξ)$µ ∈ Γo(µ, ξ#µ). Conversely if a subdifferential ξ ∈ ∂ϕ(µ) satisfies(i, ξ)$µ ∈ Γo(µ, ξ#µ), then (i, i + hξ)#µ ∈ Γo(µ, (i + hξ)#µ) for each h > 0, i.e.the curve 0 6 h 7→ (i+ hξ)#µ is a geodesic line in P2(Rd).

In spite of the fact that ’the most of’ geodesics in P2(Rd) can not be extendedto longer geodesics, the resolvents associated to functionals that satisfy (4.3.29),are defined for all h > 0 and on the whole space P2(Rd) if λ > 0. Therefore itseems interesting to understand whether there are other characterisations of theset of sufficient conditions for the construction of solutions of the Cauchy problemof Definition 4.4.1. As there are only few classes of functionals that satisfy theconditions in (4.3.29) that are known at the present time, it may as well be so thatthere is something essential about these issues, which we still do not understand.

It is natural that our semigroup of solutions of the Cauchy problem of Definition4.4.1, will be defined on D(A) by means of the exponential formula as in (4.2.14),and the minimal set of conditions required for this program is that resolventsare well defined on D(A) and for sufficiently small h > 0. We could weaken ourassumption in 1 of Definition 4.4.10 by replacing D(A) by D(A) there. In this casethe exponential formula can be proven to hold for initial points µ ∈ D(A) andsince this gives a λ-contractive semigroup on D(A), we can extend this semigroupuniquely to a λ-contractive semigroup on D(A). However we choose to have theassumption as it is currently stated.

Furthermore 2 of Definition 4.4.10 will be needed to show that the limit ofthe exponential formula actually satisfies (4.4.7). In the Hilbert space settingProposition 4.2.6 and Remark 4.2.7 show that a maximal monotone operator isstrong-weak closed in H×H. Moreover subdifferentials of functionals as in (4.3.29)have such a property as well (recall property C in the previous section). Wesimply assume our λ-MMGR operator A to be strong-weak closed as stated in 2of Definition 4.4.10, but we emphasis that it would be interesting to understandto what extent this assumption can be relaxed.

131


space P2(Rd)

4.5 AGS Subdifferentials of regular λ-convex func-tionals in generalized sense as λ-MMGR op-erators

Throughout this section ϕ is a functional defined on P2(Rd) which satisfies theconditions stated in (4.3.29) for an arbitrary but fixed constant λ ∈ R.

Our theory of λ-monotone operators on P2(Rd) should be an extension of thetheory of gradient flows presented in [5]. Therefore we need to associate a λ-MMGR operator Aϕ to ϕ, for each ϕ that satisfies (4.3.29), in a canonical way.This operator will be called the special AGS subdifferential of ϕ and denoted ∂spϕ.The solutions of the Cauchy problem (see Definition 4.4.1) associated toAϕ, shouldcoincide with the solutions of the evolution variational inequality indeed. Sincewe will construct the solutions of the Cauchy problem associated to an arbitraryλ-MMGR operator by means of the exponential formula (see Theorem 4.8.4), thetwo notions of solutions will coincide, if the resolvents defined by

Jhν := argminµ∈P2(Rd)

1

2hW 2

2 (ν, µ) + ϕ(µ)

h > 0, ν ∈ P2(Rd), (4.5.1)

coincide with resolvents associated to Aϕ according to Definition 4.4.9. Accordingto [5] Theorem 11.1.3, each solution 0 6 t 7→ µt of the evolution variational inequal-ity possesses the property that its tangent derivative (t, x) 7→ vt(x) ∈ TanµtP2(Rd)satisfies vt ∈ ∂oϕ(µt) for L1-a.e. t ∈ (0,+∞). Therefore the canonical subsetAϕ ⊂ ∂ϕ that we need to determine, should moreover satisfy

D(Aϕ) = D(∂ϕ), and ∂oϕ(µ) ∈ Aϕ(µ) ∀µ ∈ D(∂ϕ), (4.5.2)

where we follow the notation of [5] (and [17]), denoting ∂oϕ(µ) to be the uniqueelement of the minimal norm in ∂ϕ(µ) ⊂ L2(µ; Rd), for each µ ∈ D(∂ϕ). In theHilbert space setting, the canonical operator associated to a proper, convex, lowersemi-continuous functional, is just its Frechet subdifferential. However, in oursetting, it is not clear whether the AGS subdifferential ∂ϕ satisfies our conditionof monotonicity in generalized sense, for an arbitrary functional ϕ that satisfies(4.3.29). Thus we need to be more cautious. We will define a suitable subset of∂ϕ, and subsequently show that this subset has all the required properties. Let usstart with a definition.

Definition 4.5.1. Let ϕ satisfy (4.3.29). Then for µ1 ∈ D(∂ϕ) = D(|∂ϕ|) we saythat ξ1 ∈ ∂ϕ(µ1) is a special AGS subdifferential of ϕ at µ1, if for each µ2 ∈ D(ϕ)and for each ν ∈ P2(Rd) there is a 3-plan γ ∈ P2(R3d) that connects µ1 with µ2

through ν and such that

ϕ(µ2)− ϕ(µ1) >∫

Rd〈ξ(x1), x2 − x1〉dγ +

λ

2W 2γ (µ1, µ2). (4.5.3)

∂spϕ(µ1) will denote the set of vectors ξ ∈ ∂ϕ(µ1) such that (4.5.3) holds andD(∂spϕ) denotes the set of points µ ∈ D(∂ϕ) such that ∂spϕ(µ) 6= ∅. Finally the

132

Section 4.5

set of pairs [µ, ξ] ∈ ∂ϕ ⊂ P2(Rd)× L2(Rd) such that ξ ∈ ∂spϕ(µ) will be denotedby ∂spϕ. We will also be using the shorter term ’special subdifferential’ instead of’special AGS subdifferential’ sometimes.

Well the canonical operator Aϕ associated to ϕ, is the special AGS subdiffer-ential ∂spϕ. The remainder of this section is devoted to proving that ∂spϕ of ϕindeed possesses the required properties.

It is not hard to see that Definition 4.5.1 implies that the special AGS subdif-ferential ∂spϕ of ϕ is λ-MGR operator.

Lemma 4.5.2. Let ϕ satisfy (4.3.29) for some λ ∈ R. Then its special AGSsubdifferential ∂spϕ is an λ-MGR operator.

Proof. Let [µ1, ξ1], [µ2, ξ2] ∈ ∂spϕ and let ν1, ν2 ∈ P2(Rd). By Definition 4.5.1there are 3-plans γ1, γ2 ∈ P2(R3d), that connect µ1 with µ2 through ν1, ν2 respec-tively, such that

ϕ(µ2)− ϕ(µ1) >∫

Rd〈ξ1(x1), x2 − x1〉dγ1 +

λ

2W 2γ1

(µ1, µ2),

and

ϕ(µ1)− ϕ(µ2) >∫

Rd〈ξ2(x2), x1 − x2〉dγ2 +

λ

2W 2γ2

(µ2, µ1),

hold. Adding these two inequalities gives that the inequality (4.4.17) in Definition4.4.5 holds, and we conclude the claim.

In light of Lemma 4.5.2 and Lemma 4.4.8, the resolvents (recall Definition4.4.9) associated to ∂spϕ as a λ-MGR operator are unique if defined. We willdenote these resolvents by J ∂ϕh µ (for those h > 0 and µ ∈ P2(Rd) for which theyare defined).

Corollary 4.5.3. For each [µ1, ξ1], [µ2, ξ2] ∈ ∂spϕ and for each ν ∈ P2(Rd) thereis a 3-plan γ which connects µ1 with µ2 through ν such that∫

R3d〈ξ1(x1)− ξ2(x2), x2 − x1〉dγ + λW 2

γ (µ1, µ2) 6 0. (4.5.4)

Proof. Follows by Proposition 4.4.7 and Lemma 4.5.2.

Next we show that ∂spϕ is actually a λ-MMGR operator. Recall the notationintroduced in Definition 4.3.1.

Proposition 4.5.4. Suppose that ϕ satisfies (4.3.29) for some λ ∈ R. Then

∂stϕ ⊂ ∂spϕ ⊂ ∂ϕ (4.5.5)

holds.

133


space P2(Rd)

Proof. The inclusion ∂spϕ ⊂ ∂ϕ holds simply by the definition of ∂spϕ hencewe only need to show the first inclusion in (4.5.5). Let [µ0, ξ0] ∈ ∂stϕ. Thenby Definition 4.3.1 for each µ1 ∈ D(ϕ), and for each r ∈ L2(µ0; Rd) such thatr#µ0 = µ1, we have that

ϕ(µ1)− ϕ(µ0) >∫

Rd〈ξ0, r − i〉dµ0 + o(|r − i|L2(µ0;Rd)). (4.5.6)

In order to show that [µ0, ξ0] ∈ ∂spϕ according to Definition 4.5.1 fix arbitrary µ1 ∈D(ϕ) and let us consider the special case where ν ∈ Pr2 (Rd) first. Then by Lemma4.4.3 there is exactly one 3-plan which connects µ1 with µ2 through ν namely the3-plan γ := (i, rµ0

ν , rµ1ν )#ν. Moreover [0, 1] 3 t 7→ µ(t) := ((1 − t)π1 + tπ2)#γ is

the only generalized geodesic joining µ0 and µ1 with ν as the base measure, thusas ϕ is assumed to be λ-convex along generalized geodesics, ϕ must be λ-convexalong µ(t), which yields

ϕ(µ(t)) 6 (1− t)ϕ(µ0) + tϕ(µ1)− λ

2t(1− t) |rµ0

ν − rµ1ν |

2L2(ν;Rd) < +∞, ∀t ∈ [0, 1]

(4.5.7)and in particular µ(t) ∈ D(ϕ) for t ∈ [0, 1]. For t ∈ (0, 1] we can rewrite thisinequality to obtain

ϕ(µ(t))− ϕ(µ0)t

6 ϕ(µ1)− ϕ(µ0)− λ

2(1− t) |rµ0

ν − rµ1ν |

2L2(ν;Rd) . (4.5.8)

Denote rt := (1 − t)rµ0ν + trµ1

ν for t ∈ [0, 1] and observe that (rt rνµ0)#µ0 =

(rt)#ν = µ(t) for each t ∈ [0, 1] so that (4.5.6) gives∫〈ξ0, rt rνµ0

− t〉dµ0 + o(‖rt rνµ0

− i‖L2(µ0;Rd)

)6 ϕ(µ(t))− ϕ(µ0) (4.5.9)

for t ∈ [0, 1]. Furthermore

‖rt rνµ0− i‖2L2(µ0;Rd) =

∫|rt − rµ0

ν |2 dν = t2 |rµ1ν − rµ0

ν |2L2(ν;Rd) (4.5.10)

and also∫〈ξ0, rt rνµ0

− i〉dµ0 = t

∫〈ξ0(x1), x2 − x1〉dγ(x0, x1, x2)〉 (4.5.11)

hold for each t ∈ (0, 1]. Now (4.5.9), (4.5.10) and (4.5.11) yield

t

∫〈ξ0(x1), x2 − x1〉dγ(x0, x1, x2) + o(t |rµ1

ν − rµ0ν |L2(ν;Rd))

6ϕ(µ(t))− ϕ(µ0)(4.5.12)

134

Section 4.5

At last we divide inequality (4.5.12) by the quantity t |rµ1ν − rµ0

ν |L2(ν;R) = t|x2 −x1|L2(γ;R3d), and in light of (4.5.8) we obtain that for t ∈ (0, 1]∫

〈ξ0(x1), x2 − x1〉dγ|x2 − x1|L2(γ;R3d)

+o(t |rµ1

ν − rµ0ν |L2(ν;Rd))

t |rµ1ν − rµ0

ν |L2(ν;Rd)

6ϕ(µ1)− ϕ(µ0)|x2 − x1|L2(γ;R3d)

− λ

2(1− t)|x2 − x1|L2(γ;R3d).

(4.5.13)

Taking the limit for t ↓ 0 in (4.5.13) gives (4.5.3) for arbitrary choice of ν ∈ Pr2 (Rd).For general ν ∈ P2(Rd) we fix a sequence Pr2 (Rd) 3 νn → ν and finish the

proof arguing exactly as in Proposition 4.4.7.

Recall Remark 4.2.4, and observe that Lemma 4.3.4 gives a partial analogywith it. Namely, for each µ ∈ P2(Rd), and for each h > 0, the resolvent µh := Jhµsatisfies (4.3.20). However, if we are to define the resolvents by means of Definition4.4.9, then for each µ ∈ P2(Rd), and h > 0, Jhµ should be the only point ν ∈P2(Rd) such that [ν, r

µν−ih ] ∈ ∂spϕ(ν). With the aid of the results that we proved

so far, we settle this uniqueness in the following proposition. Moreover, we settlethe compatibility of the definition of the resolvents given in Definition 4.4.9 whichare denoted by J ∂ϕh , with the notion of the resolvents given in (4.5.1) which are(still) denoted by Jh.

Proposition 4.5.5. Let ϕ satisfy the conditions in (4.3.29) for some λ ∈ R. Letmoreover µ ∈ P2(Rd) and h > 0 satisfy 1 + hλ > 0. Then Jhµ is the uniquemeasure µh ∈ D(|∂ϕ|) which satisfies

ξh :=rµµh − ih

∈ ∂spϕ(µh) (4.5.14)

we have Jhµ = µh. Consequently J ∂ϕh is defined on the whole space P2(Rd), forall 0 < h < 1

λ− , and we have

Jhµ = J ∂ϕh µ , ∀µ ∈ P2(Rd), ∀0 < h <1λ−

(4.5.15)

(λ− denotes the negative part of λ).

Proof. By Lemma 4.3.4 and Proposition 4.5.4 µh := Jhµ (see 4.5.1 for definitionof Jh) satisfies (4.5.14). On the other hand by Lemma 4.5.2 and Lemma 4.4.8there can be only one such point, and we have proved the first claim. Moreover,these facts imply that the two definitions of the resolvents Jh and J ∂ϕh coincide,and than since Jhµ is defined for each µ ∈ P2(Rd), and for all 0 < h < 1

λ− (seeChapter I), we have completed the proof.

As we already mentioned, for constructing the solutions of the abstract Cauchyproblem, it is required that Aϕ = ∂spϕ must posses both properties in (4.5.2).In the following proposition we show that for each µ ∈ D(∂ϕ), the element of

135


space P2(Rd)

minimal norm ∂oϕ(µ) ∈ ∂ϕ(µ) is a special subdifferential, which settles bothclaims in (4.5.2) (see also [5] Theorem 10.3.10, for a weaker claim regarding theconvergence, under more general assumptions on ϕ).

Proposition 4.5.6. Suppose that ϕ satisfies (4.3.29). Then for each µ ∈ D(|∂ϕ|)= D(∂ϕ) the element of the minimal norm ξ0 = ∂0ϕ(µ) of ∂ϕ(µ) is a special

subdifferential of ϕ at µ, and the vectors ξh :=rµJhµ

−ih converge strongly to ξ0 as

h ↓ 0. Consequently,D(∂spϕ) = D(∂ϕ). (4.5.16)

Proof. Let µ ∈ D(|∂ϕ|) be arbitrary, and fix this µ throughout the proof. Accord-ing to Lemma 4.3.7, we have that D(|∂ϕ|) = D(∂ϕ), and there is a (unique) vectorin ∂ϕ(µ) of the minimal norm, which we denote by ξ0 := ∂oϕ(µ). Moreover, bythe same lemma, we have that

|∂ϕ|(µ) = |ξ0|L2(µ;Rd). (4.5.17)

Denote moreover µh := Jhµ, and ξh :=rµµh−ih for h > 0 such that 1 + hλ > 0.

By Proposition 4.5.5 we have that ξh ∈ ∂spϕ(µh) for each h ∈ (0, 1λ− , where λ−

denotes the negative part of the real number λ. By (3.1.20) in [5], we moreoverhave that for each h ∈ (0, 1

λ− ) that

(1 + λh)|∂ϕ|2(µh) 6(1 + λh)W 2

2 (µh, µ)h2

= (1 + λh)∫|ξh|2 dµh

61

1 + λh|∂ϕ|2(µ),

(4.5.18)

and taking lim suph↓0 in (4.5.18), together with identity (4.5.17) gives

lim suph↓0

∫|ξh|2 dµh 6 |∂ϕ|2(µ) =

∫|ξ0|2 dµ. (4.5.19)

Hence the set of plans γh := (i×ξh)#µh|h ∈ (0, 1λ− ) is weakly relatively compact

in P(R2d). Thus for any sequence hn ↓ 0, there is a subsequence (hnk)k and a2-plan γ ∈ P(R2d), such that γhnk → γ in P(R2d). Since µhn → µ in P2(Rd), wehave that (π0

2)#γ = µ. Notice that by [5] Theorem 5.4.4 (ii), we have that

ξ(x) :=∫

Rdy dγx(y), dγ(x, y) = dγx(y) dµ(x), (4.5.20)

i.e. ξ is the barycenter of the disintegration of γ with respect to µ = (π02)#γ, and

ξhnk → ξ as k → +∞, (4.5.21)

weakly, in the sense of [5] Definition 5.4.3. Well now by [5] (5.4.13), we have that

|ξ|L2(µ;Rd) 6 lim infk→+∞

|ξnk |L2(µnk ;Rd), (4.5.22)

136

Section 4.5

hence (4.5.19) and (4.5.22) together, give that

|ξ|L2(µ;Rd) 6 |ξ0|L2(µ;Rd). (4.5.23)

By Lemma 4.3.6, the convergence µhnW2−→ µ and (4.5.21) imply that ξ ∈ ∂ϕ(µ).

Since the element of minimal norm in ∂ϕ(µ) is unique, due to (4.5.23) we musthave that ξ = ξ0. The sequence hn ↓ 0 was arbitrary, so by (4.5.19) and (4.5.22)we conclude that

ξh → ξ0 strongly, as h ↓ 0. (4.5.24)

To show that ξ0 ∈ ∂spϕ(µ), choose a µ′ ∈ D(ϕ), and ν ∈ P2(Rd), and fix asequence 0 < hn ↓ 0. Then by Lemma 4.3.4 and Proposition 4.5.4, we have that

ξhn ∈ ∂spϕ(µhn) ∀n ∈ N, (4.5.25)

hence for each n ∈ N there is a 3-plan γn ∈ P(R3d), such that∫R3d〈ξhn(x1), x2 − x1〉dγn +

λ

2W 2γn(µhn , µ

′) + ϕ(µhn) 6 ϕ(µ′). (4.5.26)

Observe that by (4.5.18), we have that

µhn → µ as n→ +∞, in P2(Rd). (4.5.27)

Now we have a similar situation as in the following Theorem 4.5.7, and arguingin similar fashion, we can obtain existence of a 3-plan γ that connects µ with µ′

through ν such that∫R3d〈ξ0(x1), x2 − x1〉dγ +

λ

2W 2γ (µ, µ′) + ϕ(µ) 6 ϕ(µ′). (4.5.28)

Thus we indeed have that ξ0 ∈ ∂spϕ(µ). Identity (4.5.16) follows directly.

Now we can prove the main result of this section.

Theorem 4.5.7. Let ϕ be a functional which satisfies the conditions in (4.3.29)for some λ ∈ R. Then its special AGS subdifferential ∂spϕ is a λc-MMGR operator.Furthermore D(∂ϕ) = D(∂spϕ) holds.

Proof. Let us prove first that ∂spϕ is a λ-MMGR operator. We already know fromLemma 4.5.2 that ∂spϕ is a λ-MGR operator. Moreover by Proposition 4.5.5 itsatisfies item 1 of Definition 4.4.10 with h0 := 1

λ− . Moreover Jhµ is defined foreach µ ∈ P2(Rd) and h ∈ (0, 1

λ− ), and not only for µ ∈ D(∂ϕsp) (see [5] Theorem4.1.2(i)). In order to show that it also satisfies item 2 of Definition 4.4.10 considera sequence ([µn, ξn])n ⊂ ∂spϕ such that µn → µ ∈ P2(Rd) and ξn → ξ ∈ L2(µ; Rd)weakly (see Definition 4.3.3). Fix moreover arbitrary ν ∈ D(ϕ), σ ∈ P2(Rd).Then since ([µn, ξn])n ⊂ ∂spϕ for each n ∈ N, by Definition 4.5.1 there is a 3-planγn ∈ P2(R3d) which connects µn with ν through σ such that∫

R3d〈ξn(x1), x2 − x1〉dγn +

λ

2W 2γn(µn, µ2) + ϕ(µn) 6 ϕ(ν). (4.5.29)

137


space P2(Rd)

The weak convergence of ξn to ξ implies that

supn

∫Rd|ξn|2 dµn < +∞ (4.5.30)

hence Lemma 4.3.7 and (4.5.30) imply that

supn|∂ϕ|(µn) 6 sup

n

∫Rd|ξn|2 dµn < +∞. (4.5.31)

ϕ is assumed to satisfy (4.3.29) hence [5] Corollary 2.4.10 guarantees thatν → |∂ϕ|(ν) is lower semi-continuous. Now since µn → µ, due to (4.5.31) and thelower semi continuity of |∂ϕ| we conclude that

|∂ϕ|(µ) 6 lim infn→+∞

|∂ϕ|(µn) 6 supn|∂ϕ|(µn) < +∞. (4.5.32)

Since D(|∂ϕ|) ⊂ D(ϕ) holds by definition of the metric slope |∂ϕ| of ϕ (seeChapter III Section 2) we also have that µ ∈ D(ϕ). Now we will complete the proofby a similar kind of arguments as in Proposition 4.4.7, but with somewhat differentdetails. Recall the notation (4.4.8) and (4.4.9) and for each n ∈ N consider themultiple plans

γn := (π0, π1, π2, ξn π1, o)#γn, γ/0n := (π1,2,3,45 )#γn, (4.5.33)

where o : Rd → Rd denotes the constant mapping o ≡ 0. For each n ∈ N,the integration variable of the measures γn and γ

/0n , will be denoted by x =

(x0, x1, x2, x3, x4) and x/0 = (x1, x2, x3, x4), respectively.

supn

∫R5d|x|2 dγn(x) < +∞ (4.5.34)

and consequently we can extract a subsequence again denoted (γn)n which weaklyconverges to some 5-plan γ in P(R5d). Since marginals of weakly convergentsequences weakly converge to the same marginal of the limit we also have

γn = (π0,1,25 )#γn −→ (π0,1,2

5 )#γ =: γ weakly in P(R3d) (4.5.35)

and

γ/0n = (π1,2,3,45 )#γn −→ (π1,2,3,4

5 )#γ =: γ/0 weakly in P(R4d). (4.5.36)

The integration variable of the measures γ and γ/0 will be denoted by x =(x0, x1, x2, x3, x4) and x/0 = (x1, x2, x3, x4), respectively. Notice that [5] Proposi-tion 7.1.3 implies that γ connects µ with ν through σ. Since µn

W2−→ µ and (4.5.35)holds, an application of Lemma 4.4.4 with νn := π1,2γn there, yields

limn→+∞

W 2γn(µn, ν) = W 2

γ (µ, ν). (4.5.37)

138

Section 4.5

Thus taking lim inf at the left side of the inequality (4.5.29) we obtain

lim infn→+∞

(∫R3d〈ξn(x1), x2 − x1〉dγn

)+λ

2W 2γ (µ, ν) + ϕ(µ)

6 lim infn→+∞


)+ lim infn→+∞

λ

2W 2γn(µn, ν) + lim inf

n→+∞ϕ(µn)

6 lim infn→+∞

(∫R3d〈ξn(x1), x2 − x1〉dγn +

λ

2W 2γn(µn, ν) + ϕ(µn)

)6 ϕ(ν).

(4.5.38)We are going to show that

limn→+∞


)=∫

R3d〈ξ(x1), x2 − x1〉dγ, (4.5.39)

which in light of (4.5.38) proves that [µ, ξ] ∈ ∂spϕ. Due to (4.5.33), (4.5.35) and(4.5.36) an application of Lemma 4.4.4 gives that∫

R3d〈ξn(x1), x2 − x1〉dγn(x) =

∫R4d〈x3, x2 − x1〉dγ/0n (x/0)

−→∫

R4d〈x3, x2 − x1〉dγ/0(x/0) =

∫R5d〈x3, x2 − x1〉dγ(x)

(4.5.40)

as n→ +∞. Moreover

γ/0n = (i2, (ξn, o))#π1,2# γn ∀n ∈ N (4.5.41)

where i2 : R2d → R2d denotes the identity mapping. Well now due to (4.5.36) andour assumption that [µn, ξn] strong-weak converges to [µ, ξ], [5] Theorem 5.4.4(ii) implies that the sequence (ξn, o) ∈ L2(π1,2

# γn; R2d) weakly converges to thebarycenter of γ/0 taken with respect to the disintegration

dγ/0(x/0) = d((π1,24 )#γ

/0)ex1,ex2(x3, x4) d(π1,24 )#γ

/0(x1, x2). (4.5.42)

Let us denote this barycenter by

(ξ1, ξ1)(x1, x2) =∫

R2d(x3, x4) d((π1,2

4 )#γ/0)ex1,ex2(x3, x4). (4.5.43)

Since in the definition of this weak convergence we are allowed to take test functionsin Cyl(R2d) (recall [5] Definition 5.1.11), taking functions of the form ψ(x1, x2) :=ψ1(x1) and ψ(x1, x2) := ψ2(x2) for ψ1, ψ2 ∈ C∞c (Rd) leads to the conclusion that

ξ1(x1) = ξ(x1), ξ2(x2) = 0 γ/0 -a.e. (4.5.44)

must hold. Employing the identities (4.5.35), (4.5.36), (4.5.43) and (4.5.44) we

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space P2(Rd)

disintegrate γ/0 and compute∫R4d〈x3, x2 − x1〉dγ/0

=∫

R2d〈∫

R2dx3 d((π1,2

4 )#γ/0)ex1,ex2(x3, x4), x2 − x1〉d(π1,2

4 )#γ/0(x1, x2)

=∫

R2d〈ξ(x1), x2 − x1〉d(π1,2

4 )#γ/0(x1, x2)

=∫

R3d〈ξ(x1), x2 − x1〉dγ(x1, x2).

(4.5.45)

At last (4.5.40) and (4.5.45) yield (4.5.39) and we have proved that ∂spϕ is a λ-MMGR operator. Since the minimization problem in (4.5.1) has a unique solutionfor each h ∈ (0, 1

λ− ) (see [5] Theorem 4.1.2) and moreover by Proposition 4.5.5these solutions coincide with our notion of the resolvents, ∂ϕsp is a λc-MMGRoperator.

Our interpretation of the AGS subdifferential as a λ-MMGR operator enablesus to show an analogue of the resolvent identity (4.2.5), which we give in thefollowing proposition. For the proof see Proposition 4.7.3.

Proposition 4.5.8. Let ϕ satisfy the conditions in (4.3.29), and let 0 < h2 6 h1.Then the following resolvent identity holds:

Jh1µ = Jh2

((h2

h1rµJh1µ

+ (1− h2

h1)i)

#

Jh1µ

), 0 < h2 6 h1, µ ∈ P2(Rd).

(4.5.46)

4.6 Convex subsets of (P2(Rd),W2) in geodesic andin generalized sense

Convex subsets of Hilbert spaces are strictly related to the lower semi-continuousconvex functionals, and such sets are basic objects in convex analysis. A naturalgeneralisation in a metric space X are geodesically convex subsets of X, which arealso closely related to geodesically convex functionals defined on X, and in lightof Section 4.5 also related to λ-MMGR operators. The connection is that the levelsets

La(ϕ) := x ∈ X : ϕ(x) 6 a, a ∈ R, (4.6.1)

associated to a geodesically convex functional ϕ : X → (−∞,+∞] are geodesicallyconvex. However in light of convexity in generalized sense and monotonicity ingeneralized sense, a stronger notion of ’convex’ subsets of P2(Rd) is appropriate.This section is devoted to presenting such a notion, and investigating its properties.

140

Section 4.6

Definition 4.6.1. A subset F ⊂ P2(Rd) is geodesically convex if for each pairof measures µ0, µ1 ∈ F there is a geodesic α : [0, 1] → P2(Rd) with α(0) = µ0,α(1) = µ1, such that α(t) ∈ F for each t ∈ [0, 1].

A subset F ⊂ P2(Rd) is convex in generalized sense (abbreviated c.g.) if foreach pair of measures µ0, µ1 ∈ F and for each base measure σ ∈ P2(Rd) there is a3-plan γ ∈ P2(R3d) that connects µ0 with µ1 through σ such that the generalizedgeodesic t 7→ µt := ((1− t)π1 + tπ2)#γ satisfies µ(t) ∈ F for each t ∈ [0, 1].

If ϕ : P2(Rd)→ (−∞,+∞] is convex along generalized geodesics, then its levelsets (see (4.6.1) and take X := P2(Rd) there) are easily seen to be c.g. sets.

Remark 4.6.2. Notions of geodesically convex and strongly geodesically convexsets in (P2(Rd),W2), and some of their properties have been discussed in [43]and in [44]. In his PhD thesis [43], N.Gigli defines a subset F ⊂ P2(Rd) tobe strongly geodesically convex if for any pair of points µ0, µ1 ∈ F and any 2-plan γ ∈ ADM(µ0, µ1) := γ ∈ P2(R2d) : (π0

2)#γ = µ0, (π12)#γ = µ1, the

interpolation curve µt := ((1− t)π02 + tπ1

2)#γ satisfies µt ∈ F for all t ∈ [0, 1]. Thisproperty is clearly stronger than our notion of generalized convex sets, i.e. anystrongly convex subset of P2(Rd) is also c.g. The property of geodesically convexsubsets in P2(Rd) is stronger then the c.g. property given in Definition 4.6.1.

Lemma 4.6.3. Let F ⊂ P2(Rd) be a closed subset. Then F is c.g. if and onlyif for each µ0, µ1 ∈ F , and for each σ ∈ Pr2 (Rd), the curve µt := ((1 − t)rµ0

σ +trµ1σ )#σ ∈ F satisfies µt ∈ F for all t ∈ [0, 1].

Proof. The ’only if’ part is trivial. To prove the ’if’ part, pick µ0, µ1 ∈ F andσ ∈ P2(Rd). Let moreover Pr2 (Rd) 3 σn → σ in P2(Rd).

Arguing as in Proposition 4.4.7 we conclude that the sequence of 3-plans γn :=(i × rµ0

σn × rµ1σn)#σ is weakly relatively compact, hence we can find a 3-plan γ ∈

P(R3d) and a subsequence of (γn)n, which we again denote by (γn)n, such that

γn → γ as n→ +∞, in P(R3d). (4.6.2)

Moreover by [5] Proposition 7.1.3, we have that γ connects µ0 with µ1 throughσ. Notice also that by Lemma 4.4.3, for each n ∈ N the 3-plan γn is the unique3-plan that connects µ0 with µ1 through σn, hence due to the assumed generalizedconvexity of F , we have that

µt,n := ((1− t)rµ0σn + trµ1

σn)#σn ∈ F ∀n ∈ N,∀t ∈ [0, 1]. (4.6.3)

For each t ∈ [0, 1] and f ∈ Cb(Rd), the function R3d 3 (x0, x1, x2) 7→ f((1 −t)x1 +tx2) is clearly bounded and continuous, therefore denoting µt := ((1−t)π1 +tπ2)#σ we have that∫

f dµt,n =∫f((1− t)x1 + tx2) dγn →

∫f((1− t)x1 + tx2) dγ

=∫f dµt as n→ +∞, ∀f ∈ Cb(Rd),

(4.6.4)

141


space P2(Rd)

which amounts to

µn,t → µt as n→ +∞, in P(Rd), ∀t ∈ [0, 1]. (4.6.5)

To conclude the proof, it suffices to show that for t ∈ [0, 1]∫|x|2 dµn,t →

∫|x|2 dµt as n→ +∞ ∀t ∈ [0, 1]. (4.6.6)

Indeed by [5] Remark 7.1.11, (4.6.6) and (4.6.5) then imply that

µt,n → µt as n→ +∞ ∀t ∈ [0, 1], (4.6.7)

and since F is assumed to be closed in P2(Rd), due to (4.6.3) we must have thatµt ∈ F for all t ∈ [0, 1], and the claim of the lemma follows.

To prove (4.6.6), we observe that by [5] Lemma 5.2.4, (4.6.2) implies∫|x|2 dµn,t =

∫|(1− t)x1 + tx2|2 dπ1,2

# γn

=(1− t)2

∫|x|2 dµ0 + t2

∫|x2|2dµ1 + 2t(1− t)

∫〈x1, x2〉dπ1,2

# γn

−→ (1− t)2

∫|x|2 dµ0 + t2

∫|x2|2dµ1 + 2t(1− t)

∫〈x1, x2〉dπ1,2

# γ

=∫|x|2 dµt, as n→ +∞, ∀t ∈ [0, 1].

Proposition 4.6.4. Let F ⊂ P2(Rd) be convex in generalized sense. Then itsclosure F in P2(Rd) is also convex in generalized sense.

Proof. By Lemma 4.6.3 it suffices to show that for any µ0, µ1 ∈ F and σ ∈ Pr2 (Rd),we have that µt := ((1− t)rµ0

σ + trµ1σ )# σ ∈ F for all t ∈ [0, 1].

To this end, choose sequences (µ0,k)k and (µ1,k)k in F , such that

µj,k → µj in P2(Rd), for j = 0, 1 , (4.6.8)

and define γk :=(i, r

µ0,kσ , r

µ1,kσ

)#

for each k ∈ N. By (4.6.8) the second momentsof the 3-plans γn are bounded in n, hence the sequence (γn)n is weakly relativelycompact. Extract a subsequence of (γn)n, again denoted (γn)n, such that for some3-plan γ we have that

γn → γ, as n→ +∞, (4.6.9)

and notice that by [5] Proposition 7.1.3, γ connects µ0 with µ1 through σ.Since we assumed that σ ∈ Pr2 (Rd), Lemma 4.4.3 implies that for each n ∈

N, the 3-plan γn is the unique 3-plan that connects µ0,n with µ1,n through σ.Therefore we have that

µt,n := ((1− t)π1 + tπ2)#σn ∈ F ∀n ∈ N. (4.6.10)

142

Section 4.6

To complete the proof, we can argue just as in the proof of Lemma 4.6.3. That is,we first observe that (4.6.9) implies that

µt,n → µt = ((1− t)π1 + tπ2)#γ in P(Rd), ∀t ∈ [0, 1]. (4.6.11)

Next, applying [5] Lemma 5.2.4, we show that the convergence of the secondmoments

∫|x|2 dµt,n →

∫|x|2 dµt holds for each t ∈ [0, 1], which by [5] Remark

7.1.11 implies that (along a subsequence)

µt,n → µt as n→ +∞, ∀t ∈ [0, 1], (4.6.12)

so thatµt ∈ F ∀t ∈ [0, 1], (4.6.13)

and the proof is complete.

Next, we discuss an analogue of the closed convex hulls of subsets of P2(Rd) ingeneralized sense. The closed convex hull clcoC of a subset C of a Banach spaceB can be defined as the intersection of all closed convex subsets of B that containC. A consequence of this definition is that for any subset C ⊂ B, its closed convexhull clcoC is the smallest closed convex subset of B that contains C. Let us followthis idea.

Lemma 4.6.5. Let Fαα∈Λ be a family of closed and convex in generalized sensesubsets of P2(Rd). Then the set F :=

⋂α∈Λ Fα is also closed and convex in

generalized sense.

Proof. F is closed by the definition of the topology, hence we only need to showthe c.g. property. Pick µ0, µ1 ∈ F , and σ ∈ Pr2 (Rd). By Lemma 4.6.3, it sufficesto show that for t ∈ [0, 1], µt := ((1− t)rµ0

σ + trµ1σ )# σ ∈ F holds. By Lemma

4.4.3 the 3-plan (i × rµ0σ × rµ1

σ )#σ is the only 3-plan that connects µ0 with µ1

through σ. Therefore we have that

µt = ((1− t)π1 + tπ2)#γ ∈ Fα ∀α ∈ Λ. (4.6.14)

Thus we have that µt ∈ F for each t ∈ [0, 1], and the proof is complete.

Definition 4.6.6. The generalised closed convex hull of a subset F ⊂ P2(Rd),denoted by gclco F , is the intersection of all closed c.g. subsets C ⊃ F of P2(Rd).In light of Lemma 4.6.5, for any subset F ⊂ P2(Rd), its generalized closed convexhull gclcoF is a closed set, which is convex in generalized sense.

Remark 4.6.7. The generalised closed convex hull of a subset F ⊂ P2(Rd) canbe constructed as follows. Define the set

F1 :=

((1− t)rµ0σ + trµ1

σ )#σ| t ∈ [0, 1], µ0, µ1 ∈ F , σ ∈ Pr2 (Rd). (4.6.15)

Define moreover for n > 2 the sets

Fn :=

((1− t)rµ0σ + trµ1

σ )#σ| t ∈ [0, 1], µ0, µ1 ∈ Fn−1, σ ∈ Pr2 (Rd), (4.6.16)

143


space P2(Rd)

and let F∞ :=⋃n Fn. We clearly have that F∞ is a closed set that contains F .

In order to see that F∞ is convex in generalised sense, take µ1, µ2 ∈ F∞, and let⋃∞k=1 Fk 3 µj,n → µj in P2(Rd), as n → +∞, for j = 0, 1. Since F∞ is closed,

by Lemma 4.6.3 it suffices to show that for arbitrary σ ∈ Pr2 (Rd) we have thatµt := ((1 − t)rµ0

σ + trµ1σ )#σ ∈ F∞ holds, for each t ∈ [0, 1]. Now by definitions

(4.6.15) and (4.6.16), for each n ∈ N there is an kn ∈ N, such that µ0,n, µ1,n ∈ Fkn .Then for each n ∈ N µt,n := ((1−t)rµ1,n

σ +trµ2,nσ )#σ ∈ Fkn+1 ⊂ F∞ holds, for each

t ∈ [0, 1]. Now we can argue similarly as in the proof of Proposition 4.6.4. Thatis, we can show that the sequence of 3-plans γkn := (i, rµ0,n

σ , rµ1,nσ )#σ converges to

the 3-plan (i, rµ0σ , rµ1

σ )#σ in P2(R3d), and that moreover we can show that for eacht ∈ [0, 1], µt,n → µt = ((1 − t)rµ0

σ + trµ1σ )#σ. Since F∞ is a closed set, µt ∈ F∞

holds as well for each t ∈ [0, 1]. Thus we may conclude that clcoF ⊂ F∞ holds.Moreover it is clear that for each n ∈ N we have that Fn ⊂ F∞, hence

F∞ = clcoF . (4.6.17)

Recall that W 22 is (−1)-convex along generalized geodesics in P2(Rd), i.e. for

any triple of points µ0, µ1, ν ∈ P2(Rd) and for any 3-plan γ ∈ P(R3d) that connectsµ0 with µ1 through ν we have the following estimate

W 22 (ν, µt) 6 (1− t)W 2

2 (ν, µ0) + tW 22 (ν, µ0)− t(1− t)W 2

γ (µ0, µ1)

6 (1− t)W 22 (ν, µ0) + tW 2

2 (ν, µ0)− t(1− t)W 22 (µ0, µ1).

(4.6.18)

With the aid of the estimate (4.6.18), we can define the nearest point projectionsonto closed c.g. subsets of P2(Rd), and also prove an equivalent characterization.

Proposition 4.6.8. Let F ⊂ P2(Rd) be closed and c.g. Then for each ν ∈ P(Rd)there is a unique point PFν ∈ F such that

W2(ν, PFν) = inf W2(ν, σ)|σ ∈ F . (4.6.19)

Moreover for any ν ∈ P2(Rd) the following two properties are equivalent:

1. µ = PFν

2. For each σ ∈ F there is a 3-plan γ ∈ P2(R3d) that connects µ and σ throughν, such that ∫

〈x0 − x1, x2 − x1〉dγ 6 0 (4.6.20)

holds.

Proof. If ν ∈ F , then PFν := ν is clearly the unique point in F which sat-isfies (4.6.19). Assume that ν ∈ P2(Rd) \ F . Define λ := infσ∈F W2(ν, σ),and notice that λ < +∞ holds. Choose a sequence (µn)n ⊂ F , such thatλ = limn→∞W2(ν, µn). Since F is assumed to be c.g., for each n,m ∈ Nthere is a 3-plan γn,m ∈ P2(R3d) that connects µn with µm through ν, such thatαn,m(t) := ((1− t)π1 + tπ2)#γn,m ∈ F for all t ∈ [0, 1]. Choose ε > 0. Then there

144

Section 4.6

is an N ∈ N such that for n,m > N , we have that W 22 (ν, µn),W 2

2 (ν, µm) 6 λ2 +ε2,and with the aid of (4.6.18) we obtain that

λ2 6 W 22 (ν, αn,m(t)) 6 (1− t)W 2

2 (ν, µn) + tW 22 (ν, µm)− t(1− t)W 2

2 (µn, µm)

6 (1− t)(λ2 + ε2) + t(λ2 + ε2)− t(1− t)W 22 (µn, µm)

(4.6.21)holds for each t ∈ [0, 1]. Hence choosing t = 1

2 in (4.6.21), we deduce thatW 2

2 (µn, µm) 6 4ε2 for n,m > N . Since P2(Rd) is complete, there is a µ ∈ P2(Rd)such that µ = limn→+∞ µn. Moreover since F is assumed to be closed, we havethat µ ∈ F , and clearly W2(ν, µ) = limnW2(ν, µn) = λ.

To prove uniqueness, suppose that µ1 ∈ P2(Rd) as also a minimizer of theexpression at the right side in (4.6.19). Then in light of (4.6.21), the sequencedefined by µ2n := µ, µ2n+1 := µ1, for n ∈ N is a Cauchy sequence, which impliesthat µ = µ1.

Next we show the equivalence between the statements 1. and 2. Choose anarbitrary ν ∈ P2(Rd) and suppose that a measure µ ∈ F has the property statedin 2. Fix a σ ∈ F and let γ be a 3-plan that connects µ and σ through ν, suchthat (4.6.20) holds. Then we have that

W 22 (ν, σ) =

∫R3d|x0 − x2|2 dγ =

∫R3d|x0 − x1|2 dγ − 2

∫R3d〈x0 − x1, x2 − x1〉dγ

+∫

R3d|x2 − x1|2 dγ >

∫R3d|x0 − x1|2 dγ = W 2

2 (ν, µ).

(4.6.22)Since σ ∈ F was arbitrary, we conclude that µ = PFν.

Conversely suppose µ = PFν and let σ ∈ F be arbitrary. Let moreover γ ∈P(R3d) be a 3-plan that connects µ and σ through ν, such that µt := ((1− t)π1 +tπ2)#γ ∈ F for each t ∈ [0, 1] (such a 3-plan exists since F is assumed to be convexin generalized sense). We are now going to prove that γ satisfies (4.6.20). Assumethe opposite, i.e. assume that

∫R3d〈x0−x1, x2−x1〉dγ = c > 0 holds. Notice that

by McCann’s cyclical monotonicity criterion for optimality of a transport plan, oneeasily verifies that for each t ∈ [0, 1] we have that (π0, (1−t)π1+tπ2)#γ ∈ Γo(ν, µt).Thus for each t ∈ (0, 1) we have∫

R3d(|x0 − x1|2 − 2t〈x0 − x1, x2 − x1〉+ t2|x2 − x1|2) dγ

=∫

R3d|x0 − ((1− t)x1 + tx2)|2 dγ

=W 22 (ν, µt) > W 2

2 (ν, γ) =∫

R3d|x0 − x1|2 dγ.

(4.6.23)

Thusc

t=

1t

∫〈x0 − x1, x2 − x1〉dγ 6

∫|x2 − x1|2 dγ ∀t ∈ (0, 1). (4.6.24)

However (4.6.24) is in contradiction with the assumption that c > 0, since thenct → +∞ as t ↓ 0. We have now finished the proof.

145


space P2(Rd)

Let us recall the notion of the τ -topology on P2(Rd) discussed in [43]. Thisnotion plays the role of the weak topology on Hilbert spaces. A sequence µn ∈P2(Rd) is said to τ -converge to µ ∈ P2(Rd) if∫

f dµn →∫f dµ ∀f ∈ Cb(Rd) and sup

n

∫|x|2dµn < +∞. (4.6.25)

The τ topology is defined as the inductive limit of the topologies (Ξnτn) whereΞn := µ ∈ P(Rd)|

∫|x|2 dµ 6 n and τn is the restriction of the weak* topology

of Cb(Rd)∗ to Ξn. In [43] (see Remark 3.2) Gigli shows that a strongly convexsubset of P2(Rd) which is W2-closed must be also τ -closed. An easy modificationof that proof yields the following.

Proposition 4.6.9. Let F ∈ P2(Rd) be a c.g. set that is W2-closed. Then F isalso τ -closed.

Proof. We can follow the proof of [43] Theorem 5.8, but instead of taking anarbitrary α ∈ ADM(γ, σ), we take a transport plan γ in this set such that γ′ :=(π0

2 , π02 + π1

2 , π02 + π2

2)#γ has the property that ((1− t)π1 + tπ2)#γ′ ∈ F for each

t ∈ [0, 1]. The c.g. property of F guarantees that there is at least one such a plan.We then have that 〈x1, x2〉 < 〈γ, σ〉 and the rest of the argument in [43] Theorem5.8 can be followed to complete the proof.

4.7 Resolvents

In this section we show several fundamental results about resolvents associated toa λ-MMGR operator, which are given in Definition 4.4.9. The semigroup of solu-tions of the Cauchy problem associated to maximal monotone operators operatorson Hilbert spaces and the gradient flow semigroups on P2(Rd) are constructed withthe aid of the associated resolvents through the exponential formula (see (4.2.14)).We will also use the resolvents and the corresponding exponential formula to con-struct the solutions of the Cauchy problem associated to our maximal monotoneoperators on P2(Rd).

Naturally, we are led by the classical results on maximal monotone operatorsgiven in Section 4.2, but our situation is considerably more complex, for two rea-sons. Firstly, the tangent spaces at various points in P2(Rd) are not isometricallyisomorphic to each other. Secondly, our assumption of monotonicity in generalizedsense exploits the (−1)-convexity of W 2

2 along generalized geodesics, which is actu-ally a property of tangent spaces, hence less direct information then monotonicityin Hilbert spaces. Consequently, our proofs are more involved than the proofs ofthe results in Section 4.2.

Throughout the remainder of this section, A denotes a λ-MMGR operator,for some λ ∈ R. One of the first claims which we are going to prove concernsgeodesic convexity of D(A) (see Theorem 4.2.3 for the result on Hilbert spaces).It would also be interesting to understand, whether D(A) possesses the strongerc.g. property, since in the special case that A = ∂spϕ, we do have this property.

146

Section 4.7

If A = ∂spϕ, where ϕ satisfies (4.3.29), then the convexity of ϕ along generalizedgeodesics implies that D(ϕ) is a c.g. set, and then by Proposition 4.6.4, D(ϕ)is also a c.g. set. By Proposition 4.5.6 D(∂spϕ) = D(∂ϕ) and by Lemma 4.3.7,D(∂ϕ) = D(|∂ϕ|) holds. In light of [5] Lemma 3.1.3, we conclude that D(∂spϕ) =D(ϕ). Well now Proposition 4.6.4 yields that D(∂spϕ) is also a c.g. set.

Theorem 4.7.1. Let A be a λ-MMGR operator, for some λ ∈ R and Jh, 0 <h < h0 the resolvents associated to A. Then for any ν ∈

⋂h<h0

D(Jh) there is aν0 ∈ D(A), such that

νh := Jhν → ν0, in P2(Rd) (4.7.1)

holds, and we moreover have that

W2(ν, ν0) 6 W2(ν, σ) ∀σ ∈ D(A). (4.7.2)

In particular,

ν ∈ D(A) =⇒ νh = Jhν → ν. (4.7.3)

Proof. Recall Proposition 4.4.7. Throughout this proof, for each σ ∈ P2(Rd), and[µ, ξ], [σ, η] ∈ A, we fix a 3-plan γ ∈ P(R3d) that connects µ with σ through σ,such that ∫

〈ξ(x1)− η(x2), x2 − x1〉dγ + λW 2γ (µ, σ) 6 0. (4.7.4)

To stress the dependence of such a 3-plan γ on σ, [µ, ξ] and [σ, η], we denote this 3-plan by γ = γ(σ, [µ, ξ], [σ, η]), or just by γ = γ(σ, µ, σ) if it is clear from the contextwhich ξ and η are considered. Fix ν ∈

⋂h<h0

D(Jh). For h ∈ (0, h0), denote

νh := Jhν, set ξh :=rννh−ih , and observe that [νh, ξh] ∈ A, for each h ∈ (0, h0), so

that the 3-plans γh := γh(σ, [νh, ξh], [σ, η]) satisfy

∫〈ξh(x1)− η(x2), x1 − x2〉dγh + λW 2

γh(νh, σ) 6 0 ∀h ∈ (0, h0). (4.7.5)

Since νh ∈ D(A) ⊂ Pr2 (Rd) by definition, we have that for any σ ∈ P2(Rd), thereis only one geodesic thats connects γh and νh and ν. Therefore, if σ := ν, wemust have that ξh(x1) = x0−x1

h , γh = γh(ν, [νh, ξh], [σ, η])-a.e., for any choice of[σ, η] ∈ A, and h ∈ (0, h0). Rearranging (4.7.5), and applying the Cauchy-Schwarz

147


space P2(Rd)

inequality gives that

∫|x1|2 dνh =

∫|x1|2 dγh 6

∫〈x1, x2〉dγh +

∫〈x0, x1〉dγh −

∫〈x0, x2〉dγh

+h∫〈η(x2), x2〉dγh − h

∫〈η(x2), x1〉dγh + λh

∫|x1 − x2|2 dγh

6

√∫|x2|2 dσ

√∫|x1|2 dνh +

√∫|x0|2 dν

√∫|x1|2 dνh

+

√∫|x0|2 dν

√∫|x2|2 dσ + h

∫〈η, i〉dσ + h

√∫|η|2 dσ

√∫|x1|2 dνh

+2λh∫|x1|2 dνh = 2λh

∫|x2|2 dσ

(4.7.6)holds, for each h ∈ (0, h0). The expression at the right hand side of (4.7.6) is a

liner function in√∫|x1|2 dνh with coefficients bounded in h, so

sup∫|x1|2dγh : h ∈ (0, h0) < +∞. (4.7.7)

Hence each sequence hn ↓ 0 has a subsequence, again denoted (hn)n, such that forsome ν0 ∈ P2(Rd),

νhn → ν0, with respect to the τ topology, (4.7.8)

i.e. it converges weakly, while the second moments of (νhn)n are bounded in n.Let such a τ convergent sequence (νhn)n, together with its limit ν0 be fixed forthe moment.

Fix σ := ν throughout the remainder of the proof, and observe that for each[σ, η] ∈ A, due to (4.7.8) the second moments of the sequence of 3-plans γhn =γhn(ν, [νhn , ξhn ], [σ, η]) are bounded in n. Therefore, there is a subsequence of(hn)n, which we again denote by (hn)n, and a 3-plan γ0 ∈ P2(Rd), such that thesequence γhn τ converges to some 3-plan γ0 (the τ convergence here being definedon the set P(R3d), by slight abuse of the notation). Due to [5] Proposition 7.1.3,we have that γ0 connects ν0 with σ, through ν. In order to stress the dependenceof γ0 on σ (and he choice σ = ν), we denote this limit 3-plan by γ0 = γ0(ν, ν0, σ)(however, we do not claim that ν0 ∈ D(A)).

Recall that the functional µ 7→∫|x|2 dµ is lower semi-continuous with respect

148

Section 4.7

to the weak convergence (see [5] (7.1.10)). Due to (4.7.4) and (4.7.8) we estimate∫|x1|2 dν0 6 lim inf

n→∞

∫|x1|2 dνhn 6 lim sup

n→∞

∫|x1|2 dνhn

6 lim supn→∞

∫〈x1, x2〉dγhn + lim sup

n→∞

∫〈x0, x1〉dγhn

+ lim supn→∞

(−∫〈x0, x2〉dγhn

)+ lim sup

n→+∞λhn

∫|x1 − x2|2 dγhn

=∫〈x1, x2〉dγ0 +

∫〈x0, x1〉dγ0 −

∫〈x0, x2〉dγ0,

(4.7.9)

for arbitrary [σ, η] ∈ A (γhn = γhn(ν, [νhn , ξhn ], [σ, η]), γ0 = γ0(ν, ν0, σ)). Theequality in (4.7.9) follows by [5] Lemma 5.2.4 and the fact that the second momentsof (γhn)n are bounded in n, so that λhn lim supn→+∞

∫|x1 − x2 dγhn = 0.

So far we have obtained that for each [σ, η] ∈ A, there is a 3-plan γ0 =γ0(ν, ν0, σ), that connects ν0 with σ through ν, and such that (4.7.9) holds. Nowlet σ ∈ P2(Rd) be such that

∃(σk)k ⊂ D(A), limσk = σ with respect to τ. (4.7.10)

Then, due to (4.7.9), for each σ ∈ P2(Rd), and for each k ∈ N, there is a 3-planγ0,k = γ0,k(ν, ν0, σk), such that∫

|x1|2 dν0 6 lim infn→∞


n→∞

∫|x1|2 dνhn

6∫〈x1, x2〉dγ0,k +

∫〈x0, x1〉dγ0,k −

∫〈x0, x2〉dγ0,k

(4.7.11)

holds, for each k ∈ N. Then, since the second moments of γ0,k are boundedin k, there is a 3-plan γ0 ∈ P(R3d) such that along some subsequence (againdenoted (γ0,k)k), we have that γ0k → γ0, with respect to the topology τ . By [5]Proposition 7.1.3, we have that γ0 connects ν0 with σ through ν, and (4.7.9) stillholds. Furthermore, the inequality between the first and the last expression in(4.7.9) with γ0 replaced by γ0,k is preserved by the limit due to [5] Lemma 5.2.4.We have thus established the following:

∀σ ∈ D(A)τ∃γ0 = γ0(ν, ν0, σ) such that∫

|x1|2 dν0 6 lim infn→+∞


n→+∞

∫|x1|2 dνhn

6∫〈x1, x2〉dγ0 +

∫〈x0, x1〉dγ0 −

∫〈x0, x2〉dγ0.

(4.7.12)

Recall that νhn converges ν0 with respect to τ , thus we are allowed to chooseσ := ν0 in (4.7.12). The corresponding 3-plan γ0 = γ0(ν, ν0, ν0) satisfies π0,j

# γo ∈Γ0(ν, ν0), for j = 1, 2, which by the definition of W2 gives that∫

〈x0, x1〉dγ0 =12

(∫|x0|2 dν +

∫|x1|2 dν0 −W 2

2 (ν, ν0))

=∫〈x0, x2〉dγ0.

(4.7.13)

149


space P2(Rd)

Now by the Cauchy-Schwarz inequality, (4.7.12), and (4.7.13), we obtain the fol-lowing inequalities∫|x1|2 dν0 6 lim inf

n


n

∫|x1|2 dνhn 6

∫|x1|2 dν0. (4.7.14)

Now (4.7.8) and (4.7.14) together, give that

νhn → ν0 as n→ +∞, in P2(Rd). (4.7.15)

In particular, we have that ν0 ∈ D(A).Next, rewrite (4.7.12) in the following way:∫〈x0 − x1, x2 − x1〉dγ0 6 0, σ := ν, σ ∈ D(A)

τ, γ0 = γ0(ν, ν0, σ). (4.7.16)

Recall moreover that γ0 = γ0(ν, ν0, σ) connects ν0 with σ through ν, so that(4.7.16) gives

W 22 (ν, σ) =

∫|x0 − x2|2 dγ0

=∫ (|x0 − x1|2 + |x2 − x1|2 − 2〈x0 − x1, x2 − x1〉

)dγ0

>∫|x0 − x1|2 dγ0 = W 2

2 (ν, ν0) ∀σ ∈ D(A)τ.

(4.7.17)

Clearly, (4.7.17) implies that

W2(ν, σ) > W2(ν, ν0) ∀σ ∈ D(A). (4.7.18)

Now suppose that ν ∈ D(A). Then (4.7.18) implies that ν0 = ν must hold,and we then have that for each sequence hn ↓ 0, there is a subsequence (hnk)k,such that Jhnk ν = νhnk → ν. Therefore

νh = Jhν → ν as h ↓ 0. (4.7.19)

Suppose that ν 6∈ D(A), and suppose moreover that there are two sequenceshn, h

′n ↓ 0, such that sequences νhn = Jhnν, and νh′n = Jh′nν, have limits ν0 :=

limn νhn and ν′0 := limn νh′n . By our discussion so far, ν0, ν′0 ∈ D(A) must hold,

and in light of (4.7.12) (or rather (4.7.16)), there are 3-plans γ0 = γ0(ν, ν0, ν′0) and

γ′0 = γ′0(ν, ν′0, ν0) respectively (recall that this means that γ0 connects ν0 with ν′0through ν, and that γ′0 connects ν′0 with ν0 through ν), such that∫

〈x0 − x1, x2 − x1〉dγ0 6 0,∫〈x0 − x1, x2 − x1〉dγ′0 6 0.

(4.7.20)

150

Section 4.7

To complete the proof, we introduce the following notation. For µ1, µ2 ∈ P2(Rd)we denote 〈µ1, µ2〉o := 1

2 (∫|x1|2 dµ1 +

∫|x2|2 dµ2 −W 2

2 (µ1, µ2)). Denote for con-venience furthermore 〈π1, π2〉γ :=

∫〈x1, x2〉dγ for γ ∈ P2(R2d). The definition of

W2 implies directly that

〈µ1, µ2〉o > 〈π1, π2〉γ , ∀µ1, µ2 ∈ P2(Rd), ∀γ ∈ Γ(µ1, µ2). (4.7.21)

Now (4.7.20) and (4.7.21) give∫|x1|2 dν0 − 〈ν0, ν

′0〉o 6|π1|2γ0

− 〈π1, π2〉π1,2# γ0

6 −〈ν, ν′0〉o + 〈ν, ν0〉o∫|x1|2 dν′0 − 〈ν0, ν

′0〉o 6|π2|2γ0

− 〈π1, π2〉π1,2# γ′0

6 −〈ν, ν0〉o + 〈ν, ν′0〉o ,(4.7.22)

and by adding the inequalities in (4.7.22) we obtain

W 22 (ν0, ν

′0) = W 2

2 (ν0, σ0) +W 22 (ν′0, σ0)− 2〈ν0, ν

′0〉o 6 0,

which amounts to ν0 = ν′0. Hence, there can be only one limit of νh = Jhν ash ↓ 0, and we conclude that

Jhν → ν0 ∈ D(A), as h ↓ 0.

We have completed the proof now.

Theorem 4.7.2. Let A be a λ-MMGR operator, with λ = 0. Then D(A) is agoedesically convex set.

Proof. Consider σ0, σ1 ∈ D(A), and define

ν(t) := ((1− t)i+ trσ1σ0

)#σ0, (4.7.23)

for t ∈ [0, 1]. Clearly, t 7→ ν(t) is a geodesic. Notice that since σ1 ∈ D(A) ⊂Pr2 (Rd), the optimal transport map rσ0

σ1pushing σ1 to σ0 exists. Let us fix t ∈ (0, 1)

for the moment, and show that ν := ν(t) ∈ D(A) holds. Let Jh, 0 < h < h0 be

the resolvents associated to A. Denote νh := Jhν, ξh :=rννh−ih , for h < h0. Fix

moreover an ηj ∈ A(σj), for j = 0, 1. Such vectors exist, since σj ∈ D(A), forj = 0, 1. Recall the 3-plans γh(ν, [νh, ξh], [σj , ηj ]) =: γjh, for j = 0, 1, defined atthe beginning of the proof of Theorem 4.7.1. Since ν lies on the geodesic betweentwo regular measures σ0 and σ1, ν ∈ Pr2 (Rd) holds as well, and we have thatγjh = (i, rνhν , r

σjν )#ν, for j = 0, 1. Define the 4-plan

γh := (i, rνhν , rσ0ν , r

σ1ν )#ν, (4.7.24)

and notice that(π0,1,2

4 )#γh = γ0h, (π0,1,3

4 )#γh = γ1h. (4.7.25)

Due to Theorem 4.7.1, there is a ν0 ∈ D(A), such that νh → ν0 as h ↓ 0. Thisimplies that the family (γh)h has bounded second moments as h ↓ 0 (three d-dimensional marginals are constant in h, and the fourth one converges in P2(Rd)),

151


space P2(Rd)

hence it is weakly relatively compact in P(R4d). Hence there is a 4-plan γ0 ∈P(R4d), such that along some sequence hn ↓ 0,

γhn → γ0 as n→ +∞, in P(R4d). (4.7.26)

Letγ0

0 := (π0,1,24 )#γ0, γ1

0 := (π0,1,34 )#γ0, (4.7.27)

and observe that due to [5] Proposition 7.1.3, (π0,j4 )#γ0 is an optimal transport

plan, for j = 1, 2, 3 (recall (4.7.24)). Observe moreover that νhnW2−→ ν0 ∈ D(A)

gives that(π1

4)#γ0 = ν0. (4.7.28)

Furthermore, [5] Proposition 7.1.3 yields that

(π0,24 )#γ0 ∈ Γo(ν, σ0), (π0,3

4 )#γ0 ∈ Γo(ν, σ1), (4.7.29)

which amounts to that (π0,2,44 )#γ0 connects σ0 with σ1 through ν. Since ν is a

point on a geodesic joining σ0 with σ1, arguing as in the proof of [5] Lemma 7.2.1,with µ0 := σ0, µ1 := σ1, and µt := ν, but taking λ := (π0,2,4

4 )#γ0 there, weconclude that

x0 = (1− t)x2 + tx3 γ0-a.e. (4.7.30)

Furthermore, exactly the same arguments as in the proof of Theorem 4.7.1 inthe lines above (4.7.12) yield that∫

〈x0 − x1, x2 − x1〉dγ0,

∫〈x0 − x1, x3 − x1〉dγ0 6 0. (4.7.31)

Now multiply the first inequality in (4.7.31) by 1− t, the second one by t, and addthe two inequalities to obtain

W 22 (ν, ν0) 6

∫〈x0 − x1, x0 − x1〉dγ0 6 0. (4.7.32)

Thus (4.7.32) and Theorem 4.7.1 yield ν = ν0 ∈ D(A).Let now σ0, σ1 ∈ D(A), and take two sequences σ0,kk, σ1,k in D(A) con-

verging to σ0, and σ1, respectively, in P2(Rd). Then the sequence of 2-plansγk := (i, rσ1,k

σ0,k )#σ0,k has weak limit points, and for any such limit point γ′0 and aweakly convergent subsequence γkr → γ0 in P(R2d), we have that for t ∈ [0, 1],νt,k := ((1−t)π0 +tπ2)#γkr → ((1−t)π0 +tπ2)#γ0 =: ν(t) weakly in P(Rd) (sincethe function R2d 3 (x1, x2) 7→ (1− t)x1 + tx2 ∈ Rd is continuous). Moreover, theconvergence σj,k → σj in P2(Rd) for j = 0, 1, easily gives that

∫|x|2 dνt,k →∫

|x|2ν(t), as k → +∞, for each t ∈ [0, 1]. Therefore we have that νt,k → νt inP2(Rd). Now by [5] Proposition 7.1.3 γ0 ∈ Γo(σ0, σ1), so that by [5] Theorem 7.2.2the curve t 7→ ν(t) is a geodesic. By the first part of this proof, νt,k ∈ D(A) foreach k ∈ N and t ∈ [0, 1]. Therefore the P2(Rd) limit ν(t) of this sequence mustalso be a member of D(A). Thus, for an arbitrary pair of points σ0, σ1 ∈ D(A),we have found a geodesic t 7→ ν(t) that connects σ0 and σ1, whose image lies inD(A). The proof is now complete.

152

Section 4.7

Next we show that a natural analogue of the resolvent identity on Hilbert spaces(see Proposition 4.2.2), holds in our context. In order to formulate the resolventidentity, observe the following facts. By Theorem 4.7.2, D(A) is a geodesicallyconvex set. Moreover, for each ν ∈ D(A), and h < h0, Jhν ∈ D(A) ⊂ Pr2 (Rd), soα(t) := (trνJhν + (1 − t)i)#Jhν is the unique geodesic joining ν and Jhν. Hence,

we should have that for each 0 < h′ < h, α(

1− h′

h

)∈ D(A) ⊂ D(Jh′).

Proposition 4.7.3. Let A be a λ-MMGR operator with λ = 0, and letJh, 0 <h < h0 be its resolvents. Let moreover ν ∈ D(A), 0 < h′ < h < h0, and denoteα : [0, 1] → P2(Rd) be the unique geodesic with α(0) = ν, α(1) = Jhν ∈ D(A) ⊂Pr2 (Rd). Then

Jhν = Jh′(α

(1− h′

h

).

)(4.7.33)

Proof. By definition ξh :=rνJhν

−ih ∈ A(Jhν), and since r

α“

1−h′h”

Jhν =(

1− h′

h

)i +

h′

h rνJhν , we have that

1h′

(rα“

1−h′h”

Jhν − i

)= ξh ∈ A(Jhν) (4.7.34)

as well. By Theorem 4.7.2, D(A) is a geodesically convex set, thus α(

1− h′

h

)∈

D(A) ⊂ D(Jh′). By Definition 4.4.9, Jh′α(1− h′

h ) is the unique point µ ∈ P2(Rd)

such that 1h′

(rα“

1−h′h”

µ − i

)∈ A(µ), hence (4.7.34) guarantees (4.7.33).

We will also need the following simple result, which is nevertheless essential toconstruct the semigroup of the solutions of the Cauchy problem associated to aλ-MMGR operator (see Definition 4.4.1).

Lemma 4.7.4. Let A be a λ-MMGR operator for some λ ∈ R. Let moreoverν ∈ D(A), and 0 < h < h0 such that 1 + λh > 0. Then for any ξ ∈ Aν we have

W2(ν,Jhν) 6h

1 + λh|ξ|L2(ν;Rd). (4.7.35)

Proof. Fix ξ ∈ A(ν). Recall Proposition 4.4.7, and choose the base measureσ := Jhν so that the only 3-plan γ ∈ P(R3d) that connects ν with Jhν through

σ, corresponding to [ν, ξ], [Jhν,rνJhν

−ih ] ∈ A, is the 3-plan γ :=

(i, rνJhν , i

)#Jhν.

Thus by Proposition 4.4.7 we have that

0 >∫〈rνJhν(x2)− x2

h− ξ(x1), x2 − x1〉dγ + λW 2

γ (ν,Jhν)

=∫〈rνJhν(x)− x

h− ξ(rνJhν(x)), rνJhν(x)− x〉dJhν(x) + λW 2

γ (ν,Jhν),(4.7.36)

153


space P2(Rd)

hence by the Cauchy-Schwarz inequality(1h

+ λ

)W 2γ (ν,Jhν) 6

∫〈ξ(rνJhν(x), rνJhν(x)− x〉dJhν 6 ‖ξ‖L2(ν,Rd)Wγ(ν;Jhν)

(4.7.37)holds. Since Wγ(ν,Jhν) > W2(ν,Jhν), (4.7.35) follows

We need to define a quantity which is going to play the role of |∂ϕ| (recallLemma 4.3.7), and we define it as follows:

|Aν| := inf‖ξ‖L2(ν;Rd)| ξ ∈ A(ν)

, ν ∈ D(A). (4.7.38)

This quantity is needed in Proposition 4.7.5 below, and also for the constructionof the solutions of the abstract Cauchy problem (see Definition 4.4.1). By Lemma4.7.4 we have that

W2(ν,Jhν) 6h

1 + λh|Aν| for ν ∈ D(A) and h < h0, 1 + hλ > 0. (4.7.39)

Moreover, for any k ∈ N, ν ∈ D(A), and h as in (4.7.39), we have that

W2(J k+1h ν,J kh ν) 6

h

1 + λh

∥∥∥∥ 1h

(rJk−1h ν

J kh ν− i)

∥∥∥∥L2(J kh ν;Rd)

, (4.7.40)

since then 1h (rJ

k−1h ν

J kh ν− i) ∈ A(J kh ν) holds for each k ∈ N. Hence by induction, we

conclude that

W2(J k+1h ν,J kh ν) 6

h

(1 + λh)k+1|Aν|, ∀ν ∈ D(A), ∀k ∈ N. (4.7.41)

We will need (4.7.41) in Section 4.8. The following proposition may be consideredto be a counterpart of [5] 10.3.10, in the setting of λ-MMGR operators.

Proposition 4.7.5. Let A be a λ-MMGR operator for some λ ∈ R, let Jh, 0 <h < h0 be the associated resolvents, and let µ0 ∈ D(A). Then there is a uniquevector ξ0 ∈ A(µ0) such that

|ξ0|L2(µ0;Rd) = |Aµ0|, (4.7.42)

and we moreover have that for each sequence hn ↓ 0,

ξhn :=rµ0Jhnµ0

− ihn

−→ ξ0 ∈ L(µ0; Rd) as n→ +∞, (4.7.43)

strongly, in the sense of [5] Definition 5.4.3.

Proof. Due to (4.7.39) we have that

suph<h0

‖ξh‖L2(µh;Rd) 6 |Aµ0| < +∞, (4.7.44)

154

Section 4.8

where we set µh := Jhµ0, for h < h0.Let hn ↓ 0 be a sequence such that hn < h0, ∀n. By [5] Theorem 5.4.4.(i), there

is a subsequence of (hn)n, which we again denote by (hn)n, and a ξ0 ∈ L2(µ0; Rd)such that ξhn → ξ0 weakly. Since A is assumed to be a λ-MMGR operator,[µ0, ξ0] ∈ A holds. Now by (5.4.13) in [5] and (4.7.39), we have that that

‖ξ0‖L2(µ0;Rd) 6 lim infn→∞

‖ξhn‖L2(µhn ;Rd) 6 lim supn→∞

‖ξhn‖L2(µhn ;Rd)

6 |Aµ0| 6 ‖ξ0‖L2(µ0;Rd).(4.7.45)

Therefore, the convergence ξhn → ξ0 is strong, and (4.7.43) holds as well. Supposenow that ξ′0 ∈ A(µ0) also satisfies ‖ξ′0‖L2(µ0;Rd) = |Aµ0|, while ξ′0 6= ξ0. Define

ξ := ξ0+ξ′02 . By Lemma 4.7.4, we then have that

suph<h0

‖ξh‖L2(µh;Rd) 6 ‖ξ‖L2(µ0;Rd), (4.7.46)

and since L2(µ0; Rd) is a Hilbert space, we have that

‖ξ‖L2(µ0;Rd) <12‖ξ0‖L2(µ0;Rd) +

12‖ξ′0‖L2(µ0;Rd) = |Aµ0|. (4.7.47)

Since (4.7.47) is in contradiction with (4.7.45), we can not have that ξ0 6= ξ′0, andthe proof is complete.

In light of Proposition 4.7.5 we introduce the following definition.

Definition 4.7.6. Let A be a λ-MMGR operator for some λ ∈ R. For eachµ ∈ D(A, the unique element of the minimal norm in A(µ) ⊂ L2(µ; Rd) is denotedby Aoµ.

4.8 The abstract Cauchy problem and the con-struction of the semigroup

In this section we address questions regarding the abstract Cauchy problem asso-ciated to a λ-MMGR operator A, with λ ∈ R, which we posed in Definition 4.4.1).Our exposition is divided into three subsections. In Subsection 4.8.1 we provethe uniqueness of the solutions of the Cauchy problem, by proving the strongerλ-contraction property. The uniqueness result suffices to show the semigroup prop-erty, provided that the solutions are proven to exist.

The construction of the semigroup of solutions is divided into two furthersubsections.

In Subsection 4.8.2 we prove that the so called exponential formula is a welldefined object, i.e. the limit

Stµ0 := (J tn

)nµ0 as n→ +∞ (4.8.1)

155


space P2(Rd)

exists, for each µ0 ∈ D(A), and t > 0 where Jh, 0 < h < h0 are the resolventsassociated to A. We will use a Crandall-Liggett approach, i.e. we will use thetechnique of doubling of the number of variables. One of the consequences ofusing the Crandall-Liggett technique is that we will obtain convergence of order

1√n

of the discrete scheme to its limit. It is well known that the optimal order ofconvergence of a discrete scheme associated to an accreative operator on a Banachspace, and also of a discrete scheme for gradient flows on metric spaces undergeneralized convexity assumptions, is 1

n (see [81] and [5] Chapter 4, respectively).The author of this thesis intends to attempt providing an improvement of his workin this respect, in the future.

In Subsection 4.8.3 we prove that the limit of the discrete scheme indeed yieldsthe (unique) solutions of the abstract Cauchy problem. In order to prove thisfact, following the Crandall-Liggett approach, which involves a dual of a Banachspace, does not seem to be an easy way to go. Furthermore, the approach of [5],which could be seen as a discrete improved modification of the proof in [17] (seeTheoreme 3.1) would require finding a counterpart of the variational apparatusavailable for functionals, and is therefore not directly applicable in our situationeither. We will take a different approach instead, where instead of consideringpiecewise constant curves induced by the discrete time solutions as in [5], weconsider piecewise geodesic curves induced by the discrete time solutions.

Once we establish the above announced results, due to the semi-contractionproperty to be proven in Subsection 4.8.1, we can uniquely extend the semigroup ofsolutions (St)t>0 defined by (4.8.1), to a semi-contraction semigroup on D(A). Themonotonicity in generalized sense and the generalized (−1)-convexity of W 2

2 playessential role in our constructions carried out in Subsection 4.8.2, and Subsection4.8.3.

Throughout the remainder of the section, we fix a constant λ ∈ R, and a λ-MMGR operator A. Recall that in such case, there is a constant h0 > 0 such thatfor h < h0 the resolvents Jh associated to A are well defined (see Definition 4.4.9and Definition 4.4.10), and throughout the remainder of this section we fix onesuch constant h0 > 0.

4.8.1 Uniqueness and the semi-contraction property of thesolutions

Proposition 4.8.1. Let T > 0 and let µ1, µ2 : [0,+∞)→ Pr2 (Rd) be two solutionsof the abstract Cauchy problem associated to a λ-MMGR operator A. Then for0 6 t < +∞ we have that

W2(µ1t , µ

2t ) 6 e

−λt2 W2(µ1

0, µ20). (4.8.2)

In particular, if µ10 = µ2

0, then µ1t = µ2

t for all t ∈ [0,+∞).

Proof. Denote (ξjt )t∈[0,+∞) to be the (L1-a.e. on [0,+∞) defined) vector fieldcorresponding to (µjt )t for j = 0, 1, such that (4.4.5), (4.4.6), and (4.4.7) hold

156

Section 4.8

(with ξjt substituted for vt and µjt for µt, for j = 1, 2). Let δ(t, s) := W 22 (µ1

t , µ2s).

By [5] Theorem 8.4.7, we have that for each s ∈ [0,+∞)

d

dtδ(t, s) =

∫〈ξ1t (x1), x2 − x1〉dγt,s(x1, x2), ∀γt,s ∈ Γo(µ1

t , µ2s),

for L1 -a.e. t ∈ [0,+∞).(4.8.3)

Likewise, for each t ∈ [0,+∞), we have that

ddsδ(t, s) =

∫〈ξ2s (x2), x1 − x2〉dγt,s(x1, x2) ∀γt,s ∈ Γo(µ1

t , µ2s),

for L1 -a.e. s ∈ [0,+∞).(4.8.4)

In order to apply (an obvious modification of) [5] Lemma 4.3.4, fix T > 0. Letc := sups,t∈[0,T ]W

22 (µ1

s, µ2t ), which is a finite constant, since by assumption both

curves µ1t and µ2

t are assumed to be continuous, and define moreover the function

0 6 t 7→ u(t) :=∫ t

0

(12‖ξ1r‖2L2(µ1

r;Rd) +12‖ξ2r‖2L2(µ2

r;Rd) + C

)dr. (4.8.5)

By the Cauchy-Schwarz and Young inequality, for 0 6 s1 < s2 6 T , 0 6 t1 < t2 6T , and s, t ∈ [0, T ], we have that

|δ(t2, s)− δ(t1, s)| 6 |u(t2)− u(t1)|,|δ(t, s2)− δ(t, s1)| 6 |u(s2)− u(s1)|.

(4.8.6)

Since t 7→ u(t) is clearly absolutely continuous, in light of [5] Lemma 4.3.4, wehave that for L1-a.e. t ∈ [0, T ]

ddtδ(t, t) 6 lim sup

h↓0

δ(t, t)− δ(t− h, t)h

+ lim suph↓0

δ(t, t+ h)− δ(t, t)h

. (4.8.7)

Now by (4.8.3), (4.8.4), and (4.8.7), we have that for L1-a.e. t ∈ [0, T ]

ddtδ(t, t) 6 −

∫〈v1t (x1)− v2

t (x2), x1 − x2〉dγt,t(x1, x2) (4.8.8)

holds, for each γt,t ∈ Γo(µ1t , µ

2t ). By Definition 4.4.1, for L1-a.e. t ∈ [0, T ],

µjt ∈ D(A) ⊂ Pr2 (Rd), for j = 1, 2, hence for L1-a.e. t ∈ [0, T ], Γo(µ1t , µ

2t ) is a

one point set. Therefore, the right hand side in (4.8.8) is bounded from aboveby −λW 2

2 (µ1t , µ

2t ), for L1-a.e. t ∈ [0, T ], and (4.8.2) follows by the Gronwall

inequality.

Remark 4.8.2. Recall that by Remark 4.4.6, l-MGR operators are also λ-monot-one operators. Therefore, Proposition 4.8.1 applies to λ-MGR and λ-MMGRoperators as well, which is the case of primary interest in this exposition. Further-more, notice that the semi-contraction property proven in Proposition 4.8.1 below,follows from the weaker λ-monotonicity property, the stronger λ-MGR propertybeing redundant (see also Remark 4.8.2 below).

157


space P2(Rd)

4.8.2 The exponential formula - Part 1

In order to prove the main theorem of this subsection, we need the followingmodification of an auxiliary lemma from [30]. This modification is given by P.Clement in his lecture notes on gradient flows [26].

Lemma 4.8.3. Let r, h, δ, k be real numbers such that

0 < r 6 2, h, δ, k > 0. (4.8.9)

Moreover let n,m ∈ N, and let (ai,j) 0 6 i 6 m, 0 6 j 6 n be non-negative realnumbers, such that

ai,j 6h

h+ δai,j−1 +

δ

h+ δai−1,j , for 1 6 i 6 m, 1 6 j 6 n, (4.8.10)

ai,0 6 K(ih)r

for 1 6 i 6 m, (4.8.11)

a0,j 6 K(jδ)r for 1 6 j 6 n. (4.8.12)

Then for 1 6 i 6 m and 1 6 j 6 n, we have that

ai,j 6 K[(ih− jδ)2 + (h+ δ) minih, jδ]r/2. (4.8.13)

Proof. See [26] Lemma A-2.

Recall the quantity |A|(µ0), for µ0 ∈ D(A), defined in (4.7.38). Recall moreoverthat by Remark 4.4.6, if A is a λ0-MGR operator for λ0 > 0, then A is also aλ-MGR operator, for λ 6 0, and the we may clearly substitute MGR by MMGR,in this claim. Hence, there is no loss of generality in assuming that λ 6 0 4

Theorem 4.8.4. (Exponential formula) Let λ 6 0, let A be a λ-MMGR operatoron (P2(Rd),W2), and let Jh, 0 < h < h0 be the associated resolvents. Then foreach µ0 ∈ D(A) and t > 0, the limit

Stµ0 := limn→∞

(J tn

)nµ0 in P2(Rd) (4.8.14)

exists. Furthermore, for each T > 0, the curve 0 6 t 7→ Stµ0 ∈ P2(Rd) is Lipschitzon [0, T ]. Precisely, we have the following estimate

W2(Ssµ0, Stµ0) 6 |Aµ0|e−λt|t− s|, for 0 6 s 6 t < +∞. (4.8.15)

4 To picture this situation better, suppose that ϕ a functional defined on a Hilbert space H.If for λ2 > 0, x 7→ ϕ(x)− λ2|x|2 is convex, then certainly x 7→ ϕ(x)− λ1|x|2 = ϕ(x)− λ2|x|2 +(λ2 − λ1)|x|2 is convex as well, provided that λ1 6 λ2.

158

Section 4.8

Proof. Fix µ0 ∈ D(A), and t > 0. Let us first show that the iterations at the righthand side of (4.8.14) converge in P2(Rd) and that we have the following errorestimate:

W2

(Stµ0,

(J tn

)nµ0

)6 |Aµ0|C(t, n), if 1 +

λt

n> 0, n ∈ N,

where C(t, n) := t

(1n

+(λt

n

)2)1/2

max

e−λt,

(1 +

λt

n

)−(n+2) 1

2

.

(4.8.16)

To this aim, fix t > 0, and 0 < h, δ < h0 such that 1 + λh, 1 + λδ > 0.By (4.7.41), for n ∈ N we have that

W 22 (J nh µ0, µ0) 6

(n∑k=1

W2(J kh µ0,J k−1h µ0)

)2

6 n

n∑k=1

W 22 (J kh µ0,J k−1

h µ0)

6nn∑k=1

h2

(1 + λh)2k|Aµ0|2.

(4.8.17)Since λ 6 0, and h > 0 we have that 1

1+λh > 1. Hence, for each n ∈ N, and foreach k ∈ 1, ..., n, 1

(1+λh)k6 1

(1+λh)n holds, so that (4.8.17) implies that

W 22 (J nh µ0, µ0) 6

n2h2

(1 + λh)2n|Aµ0|2 ∀n ∈ N. (4.8.18)

Likewise we have that

W 22 (Jmδ µ0, µ0) 6

m2δ2

(1 + λδ)2m|Aµ0|2 ∀m ∈ N. (4.8.19)

Letµ0,h := µ0, µi,h := J ihµ0 for i ∈ N,

µ0,δ := µ0, µj,δ := J jδ µ0 for j ∈ N.(4.8.20)

Define furthermore the following non-negative real numbers

ai,j := (1 + λh)i(1 + λδ)jW 22 (µi,h, µδ,j), i, j ∈ N. (4.8.21)

By Definition 4.4.10, for each i, j ∈ N we have that

ξi,h :=1h

(rµi−1,hµi,h − i

)∈ A(µi,h), ξj,δ :=

1δ

(rµj−1,δµj,δ − i

)∈ A(µj,δ). (4.8.22)

Now, recall Definition 4.4.5, and for each i, j > 2, choose

[µ1, ξ1] := [µi,h, ξi,h], [µ2, ξ2] := [µj,δ, ξj,δ],σ1 := µi−1,h, σ2 := µj−1,δ,

γ1,h,δ,i,j := (i, rµi,hµi−1,h , rµj,δµi−1,h)#µi−1,h,

γ2,h,δ,i,j := (i, rµi,hµj−1,δ , rµj,δµj−1,δ)#µj−1,δ,

(4.8.23)

159


space P2(Rd)

in this definition. Since Jtµ0 ∈ D(A) ⊂ Pr2 (Rd) for each t > 0, due to Lemma4.4.3, γ1,h,δ,i,j is the unique 3-plan that connects µi,h with µj,δ through µi−1,h,and γ2,h,δ,i,j is the unique 3-plan that connects µi,h with µj,δ through µj−1,δ.Therefore, by (4.4.17), and (4.8.22) we have that∫

〈 1h

(x0 − x1), x2 − x1〉dγ1,h,δ,i,j +∫〈1δ

(x0 − x2), x1 − x2〉dγ2,h,δ,i,j

+λ

2W 2γ1,h,δ,i,j

(µi,h, µj,δ) +λ

2W 2γ2,h,δ,i,j

(µi,h, µj,δ) 6 0.(4.8.24)

Observe that π1,2# γ1,h,δ,i,j ∈ Γ(µi,h, µj,δ) and π1,2

# γ2,h,δ,i,j ∈ Γ(µi,h, µj,δ), for eachi, j > 2. To make the notation shorter, fix i, j > 2 for the moment, and setγ1 := γ1,h,δ,i,j , γ2 := γ2,h,δ,i,j , and observe that∫

〈x0 − x1, x2 − x1〉dγ1 =12

∫(|x0 − x1|2 + |x2 − x1|2 − |x0 − x2|2) dγ1

>12W 2

2 (µi−1,h, µi,h) +12W 2

2 (µj,δ, µi,h)− 12W 2

2 (µi−1,h, µj,δ).

(4.8.25)In a similar fashion we obtain∫

〈x0 − x2, x1 − x2〉dγ2

>12W 2

2 (µj−1,δ, µj,δ) +12W 2

2 (µi,h, µj,δ)−12W 2

2 (µj−1,δ, µi,h).(4.8.26)

Now (4.8.24), (4.8.25), and (4.8.26) imply that (we omit two positive terms at theleft hand side)

12hW 2

2

(J ihµ0,J jδ µ0

)+

12δW 2

2

(J ihµ0,J jδ µ0

)+λ

2W 2γ1,h,δ,i,j

(J ihµ0,J jδ µ0

)+λ

2W 2γ2,h,δ,i,j

(J ihµ0,J jδ µ0

)6

12hW 2

2

(J i−1h µ0,J jδ µ0

)+

12δW 2

2

(J j−1δ µ0,J ihµ0

),

(4.8.27)

hence

W 22

(J ihµ0,J jδ µ0

)6

h

h+ δ + 2λhδW 2

2

(J ihµ0,J j−1

δ µ0

)+

δ

h+ δ + 2λhδW 2

2

(J jδ µ0,J i−1

δ µ0

).

(4.8.28)

Recalling the notation (4.8.21) and setting moreover h := h(1+λδ), δ := δ(1+λh),(4.8.28) yields

ai,j 6h

h+ δai,j−1 +

δ

h+ δai−1,j . (4.8.29)

160

Section 4.8

Further, inequalities (4.8.18), (4.8.19) and the definition of h and δ, together withinequalities 1

(1+λh)i 6 1(1+λh)m for i 6 m and 1

(1+λδ)j 6 1(1+λδ)n for j 6 n imply

that

ai,0 6 |Aµ0|21

(1 + λh)m· 1

(1 + λδ)2· (ih)2, n ∈ N, i ∈ 1, · · · ,m, (4.8.30)

a0,j 6 |Aµ0|21

(1 + λδ)n· 1

(1 + λh)2· (jδ)2, n ∈ N, j ∈ 1, · · · , n. (4.8.31)

Finally since

|Aµ0|2 max(1 + λh)−m(1 + λδ)−2, (1 + λδ)−n(1 + λh)−26|Aµ0|2 max(1 + λh)−(m+2), (1 + λδ)−(n+2),

we set K := |Aµ0|2 max(1 + λh)−(n+2), (1 + λδ)−(m+2), so that by (4.8.29),(4.8.30), and (4.8.31), the conditions of Lemma 4.8.3 are satisfied, and we haveobtained

W 22 (Jmh µ0,J nδ µ0) 6|Aµ0|2 max(1 + λh)−(n+2), (1 + λδ)−(n+2)·

((mh− nδ) + (m− n)λhδ)2 + (h+ δ) minmh, nδ.(4.8.32)

Now choose n0 ∈ N such that 0 < 1 + λ tn0

holds. Then for n,m > n0, 1 + λ tn , 1 +λ tm > 1 + λt

n0> 0. Choosing h = t

m , δ = tn , (4.8.32) gives that

W2

(J ntnµ0,Jmt

mµ0

)6 |Aµ0|t

·max(1 +λt

n)−(n+2), (1 +

λt

m)−(m+2) 1

2 · ( 1n

+1m

+ (λt)2(1n− 1m

)2)1/2.

(4.8.33)

Since(1 + λt

n

)−(n+1),(1 + λt

m

)−(m+1) → e−λt, (4.8.33) implies that(J tn

)nnµ0

is a Cauchy sequence in P2(Rd), and we denote its limit by Stµ0. Thus, (4.8.14)is established. Letting m→ +∞ in (4.8.33) gives (4.8.16).

To establish (4.8.15), we choose 0 < s 6 t < +∞, and µ0 ∈ D(A). Then forn ∈ N such that 1 + λ

n > 0 and tn < h0, we choose m = n, h := t

n , δ := sn in

(4.8.32) to obtain

W 22

((J tn

)nµ0,(J sn

)nµ0

)6 |Aµ0|2

(1 +

λt

n

)−2(n+1)(t− s)2 +

t+ s

ns

(4.8.34)

for such integers n. Letting n → +∞ in (4.8.34), we obtain (4.8.15) for 0 < s 6t < +∞. Finally take h := t

n in (4.8.18), and send n→ +∞ to obtain (4.8.15) fors = 0. The proof is complete.

161


space P2(Rd)

4.8.3 The exponential formula - Part 2

In this subsection, we will show that the curves 0 6 t 7→ Stµ0, µ0 ∈ D(A), con-structed in Theorem 4.8.1, are in fact the solutions of the abstract Cauchy problemassociated to A. Our proof takes a different approach than the proofs in Hilbertspaces and for gradient flows on P(Rd) (see [17] and [5]). We will consider mea-sures on [0, t]× R3d, which are constructed with the aid of the piecewise geodesiccurves induced by the discrete time solutions (i.e. iterations of the resolvents) andtheir Wasserstein velocity fields.

In order to proceed, we need the following auxiliary lemma.

Lemma 4.8.5. Let t > 0, m ∈ N, and let moreover (γn)n∈N ⊂ P([0, t]×Rm) andγ ∈ P([0, t]× Rm), be such that

πt#γn = L1

[0,t] ∀n ∈ N, πt#γ := L1[0,t], πt : [0, t]× Rm → [0, t], πt(s, x) := s,

(4.8.35)and denote

dγn = dνns ds, n ∈ N, dγ = dνs ds (4.8.36)

to be the L1-a.e. s ∈ [0, t] defined disintegrations of γn and γ, with respect to L1[0,t]

(see [5] Theorem 5.3.1, or [36]). Suppose that

γn −→ γ as n→ +∞, in P([0, t]× Rm) (4.8.37)

holds. Thenνns → νs in P(Rm), for L1–a.e. s ∈ [0, t] (4.8.38)

holds too.

Proof. In light of [36], it is enough to show that there is a set N ∈ B([0, t]) (theσ-algebra of Borel subsets of [0, t]), such that L1(N ) = 0, and such that for eachs ∈ [0, t] \ N and n ∈ N, νns and νs are defined, and∫

f dνns →∫f dνs ∀f ∈ C0(Rm). (4.8.39)

There C0(Rm) denotes the Banach space of continuous real valued functions de-fined on Rm, that vanish at infinity. For n ∈ N, let Nn,N ∈ B([0, t]) be sets suchthat L1(Nn) = L1(N ) = 0 for each n ∈ N and such that for each s ∈ [0, t] \ Nnand s ∈ [0, t]\N , νns and νs, respectively, are well defined. Set N ′ := N ∪ (∪nNn).Next, let frr∈N be a dense sequence in C0(Rm) (recall that C0(Rm) is separable).Then for each r ∈ N and each g ∈ Cb([0, t]) we have that (s, x) 7→ g(s)fr(s, x) ∈Cb([0, t]× Rm). Therefore∫ t

0

g(s)∫

Rmfr dνns ds =

∫[0,t]×Rm

g(s)fr(x) dγn(s, x)

n→+∞−→∫

[0,t]×Rmg(s)fr(x) dγ(s, x) =

∫ t

0

g(s)∫

Rmfr(x) dνs ds,

(4.8.40)

162

Section 4.8

which implies that for r ∈ N there is a set N ′r ∈ B([0, t]), such that L1(N ′r) = 0,and such that for each s ∈ [0, t] \ (N ′r ∪N ′) we have that∫

fr dνns →∫fr dνs as n→∞. (4.8.41)

Define M := N ′ ∪ (∪rN ′r), and observe that L1(M) = 0. We conclude the proofby showing that for s ∈ M, νns → νs in P(Rm). To this aim, choose f ∈ C0(Rm)and ε > 0. Then there is a r ∈ N, such that ‖fr − f‖C0(Rm) < ε. Moreover, thereis an n0 ∈ N such that for n > n0,

∣∣∫ fr dνns −∫fr dνs

∣∣ < ε holds. Finally, weestimate∣∣∣∣∫ f dνns −

∫f dνs

∣∣∣∣ 6 ∣∣∣∣∫ f dνns −∫fr dνns

∣∣∣∣+∣∣∣∣∫ fr dνns −

∫fr dνs

∣∣∣∣++∣∣∣∣∫ fr dνs −

∫f dνs

∣∣∣∣ 6 2‖f − fr‖∞ + ε 6 2ε,

for n > n0, which completes the proof.

For a fixed t > 0, and n ∈ N, and µ0 ∈ D(A), we have that (J tn

)nµ0 ∈ D(A)holds by definition. Fix t > 0 and µ0 ∈ D(A) throughout the remainder of thissubsection, and denote for n ∈ N and for k ∈ 1, ..., 2n, rnk to be the uniqueoptimal transport map which pushes (Jt2−n)kµ0 to (Jt2−n)k−1µ0, where we define(Jh)0µ0 := µ0, for h > 0.

Due to (4.7.41), and [5] Theorem 5.4.4, the sequence ξn := rnn−it2−n has a weakly

convergent subsequence. Due to convergence in (4.8.14), and item 2 of Definition4.4.10, any weak limit point ξ of (ξn)n satisfies ξ ∈ A(Stµ0), and in particularStµ0 ∈ D(A) holds. However we wish to show that a Wasserstein derivativevector field of the curve t 7→ Stµ0 =: µt satisfies (4.4.7). Observe that sinceby Theorem 4.8.4 this curve is locally Lipschitz on [0,+∞), it is also of classAC2([a, b];P2(Rd)), for 0 6 a < b < +∞, while since there are uncountably manyt ∈ [0,+∞), a diagonal type of argument does not work here. To this aim fix t > 0and µ0 ∈ D(A). We are going to define a sequence of piecewise geodesic curves asfollows. Define firstly,

µn0 := µ0, µnkt2−n := (Jt2−n)k µ0, for k = 1, 2, ..., 2n, (4.8.42)

and denoteξnk :=

rnk − it2−n

∈ A(µnkt2−n), for k = 1, ..., 2n. (4.8.43)

For each s ∈ [0, t], let kns ∈ 0, 1, ..., 2n − 1 and 0 6 δns < 2−nt be the uniquenumbers such that

s = kns 2−nt+ δns . (4.8.44)

Define piecewise geodesic curves by

µns =((

1− δns2−nt

)rn(kns+1)t2−n +

δns2−nt

i

)#

µn(kns+1)t2−n , for s ∈ [0, t].

(4.8.45)

163


space P2(Rd)

By definition, for each n ∈ N and k ∈ 1, ..., 2n the restriction of the curve µn

to [(k − 1)t2−n, kt2−n] is a geodesic joining µn(k−1)t2−n and µnkt2−n . In order toshorten the notation somewhat, we will denote

µn,k := µnkt2−n , for k = 0, 1, ..., 2n, n ∈ N. (4.8.46)

Notice that each curve s 7→ µns is piecewise a geodesic, hence also Lipschitz on[0, t], thus also of class AC2((0, t);P2(Rd)). The piecewise geodesic property of thecurves s 7→ µns also implies that their metric derivatives s 7→ |µn|(s) satisfy

|µn|(s) ≡ ‖ξnk+1‖L2(µn,k+1;Rd), n ∈ N, k = 0, ..., 2n − 1, s ∈ (k2−nt, (k + 1)2−nt),(4.8.47)

and the same equality holds for the right and left metric derivative of µn at s =k2−nt and s = (k + 1)2−nt, respectively. Notice that (4.7.40) and (4.7.41) yieldthat(

1 +λt

2n

)2n

‖ξn2n‖L2(µn,2n ;Rd) 6

(1 +

λt

2n

)2n−1 ∥∥ξn2n−1

∥∥L2(µn,2n−1;Rd)

6 · · · 6(

1 +λt

2n

)‖ξn1 ‖L2(µn,1;Rd) 6 |Aµ0|.

(4.8.48)

We will also consider the following right continuous functions:

gn(s) := (1 + λt2−n)k+1|µn|(s) if s ∈ [k2−nt, (k + 1)2−nt), k = 0, 1, ..., 2n − 1 ,(4.8.49)

for n ∈ N. Due to (4.8.47), and (4.8.48), for each n ∈ N, the function [0, t] 3 s 7→gn(s) is non-increasing. At last, we denote

µs := Ssµ0 = limn

(J sn

)nµ0, for s ∈ [0, t], (4.8.50)

and we denote (vs)s∈[0,t] to be the tangent Wasserstein velocity field associated tos 7→ µs, i.e. the L1-a.e. s ∈ [0, t] uniquely defined Borel vector field [0, t] 3 s 7→vs ∈ TanµsP2(Rd) such that the continuity equation holds (see [5] Theorem 8.3.1).

We wish to describe the Wasserstein-2 velocity vector fields of the curves s 7→µns as well. Since [0, t] 3 s 7→ µns is piecewise geodesic, the vector field

[0, t) 3 s 7→ vns :=1

2−nt− δns

(rµn,kns +1

µns− i)∈ Tanµns P2(Rd) (4.8.51)

is a version of the L1-a.e. defined tangent velocity field of [0, t] 3 s 7→ µns , for eachn ∈ N. Moreover, for each n ∈ N, s ∈ [0, t], and the corresponding numbers knsand δns , we have that

rµnsµn,kns +1 =

(1− δns

2−nt

)rµn,knsµn,kns +1 +

δns2−nt

i. (4.8.52)

164

Section 4.8

Now choose ψ ∈ C∞c (Rd), and apply (4.8.52) to compute∫Rd〈∇ψ, vns 〉dµns =

∫ ⟨∇ψ

(rµnsµn,kns +1

), vns

(rµnsµn,k+1

)⟩dµn,kns+1

=∫

Rd

⟨∇ψ

((1− δns

2−nt


δns2−nt

i

),−ξnkns+1

⟩dµn,kns+1.

(4.8.53)

For ψ ∈ C∞c ([0, t) × Rd) (denoting ψs := ψ(s, .), for s ∈ [0, t)), the continuityequation (see [5] (8.1.4)) gives that∫ t

0

∫Rd

(∂sψ(s, x) + 〈∇xψ(s, x), vns (x)〉) dµns ds+∫

Rdψ(0, x) dµ0

=∫ t

0

∫Rd

(∂sψ(s, x) + 〈∇xψ(s, x), vs(x)〉) dµs ds+∫

Rdψ(0, x) dµ0

=0 ∀n ∈ N.

(4.8.54)

Due to Theorem 4.8.1, it is not hard to show that for each s ∈ [0, t], µnsW2→ µs

as n → +∞, and since the function ∂sψ is bounded, the dominated convergencetheorem yields that∫ t

0

∫Rd∂sψ(s, x) dµns ds −→

∫ t

0

∫Rd∂sψ(s, x) dµs ds as n→ +∞. (4.8.55)

Now (4.8.54) and (4.8.55) yield that∫ t

0

∫Rd〈∇xψ(s, x), vns (x)〉 dµns ds −→

∫ t

0

∫Rd〈∇xψ(s, x), vs(x)〉 dµs ds (4.8.56)

as n→ +∞. Recall that we have to show that −vs ∈ A(µs), for L1-a.e. s ∈ [0, t].Since for each s ∈ [0, t], we have that (kns +1)2−nt ↓ s, and due to (4.8.48), and [5]Theorem 5.4.4, each subsequence of (kns )n, has a further subsequence such that thecorresponding subsequence of −ξnkns+1 = vnkns+1 weakly converges (in the sense of[5] Definition 5.4.3) to a vector −ξs ∈ L2(µs; Rd). By item 2 of Definition 4.4.10,we then have that µs ∈ D(A), and also [µs,−ξs] ∈ A. However, we need the vectorfield (s, x) 7→ ξs(x) to be Borel, while there are uncountably many s ∈ [0, t], andwe need to perform a suitable manouvre in order to circumvent this problem.

Let us show first that the curves [0, t] 3 s 7→ µns converge uniformly to thecurve [0, t] 3 s 7→ µs.

Lemma 4.8.6. The sequence of curves s 7→ µns converges uniformly on [0, t] withrespect to W2, to the curve s 7→ µs, i.e.

limn→∞

sups∈[0,t]

W2(µns , µs) = 0. (4.8.57)

165


space P2(Rd)

Proof. For n ∈ N, define Dtn := k2−nt|k = 0, 1, ..., 2n, and let Dt =

⋃nD

tn. We

then have that Dt = [0, t]. Let m ∈ N and consider s = k2−mt ∈ Dtm. Then for

N ∈ n > m, s = kn2−nt holds, with kn := k2n−m. Clearly kn → +∞ if n → ∞,which by Theorem 4.8.1 implies that

µns = (Jt2−n)kn µ0 =(J skn

)knµ0 → Ssµ0 = µs as n→ +∞. (4.8.58)

Next, we fix s ∈ [0, t] \ Dt. In light of (4.8.15), (4.8.47) and (4.8.48), the familyof curves s 7→ µns and s 7→ µs is equi-Lipschitz on [0, t], thus also equi-continuouson [0, t]. Choose ε > 0. Then there is a δ > 0 such that if 0 6 s1 < s2 6 t ands2 − s1 < δ then W2(µs1 , µs2) < ε, and also W2(µns1 , µ

ns2) < ε for each n. Choose

moreover m ∈ N and k ∈ 0, 2, ..., 2m − 1 such that k2−mt < s < (k + 1)2−mtand 2−mt < δ.

Now we conclude two things. First, due to Theorem 4.8.1, for n large enoughwe have

W2(µnk2−mt, µk2−mt) < ε, (4.8.59)

hence for large n

W2(µns , µs) 6 W2(µns , µnk2−mt) +W2(µnk2−mt, µk2−mt) +W2(µk2−mt, µs) 6 3ε,

(4.8.60)which amounts to

µns → µs in P2(Rd), as n→ +∞. (4.8.61)

Secondly, the set of time points Dtm is finite, hence there is an n0 ∈ N such that

for n > n0 and s ∈ Dtm, we have that

W2(µns , µs) < ε. (4.8.62)

Repeating the argument as in (4.8.60), we deduce that this estimate actually holdsuniformly for all s ∈ [0, t], which gives (4.8.57).

Now we are ready to crown our efforts. Recall that since a λ-MMGR operatorfor some λ > 0, is also a λ-MMGR operator for each λ 6 0, there is no loss ofgenerality to assume that λ 6 0. Recall also that by Proposition 4.7.5, for eachµ0 ∈ D(A), the set A(µ0) ⊂ L2(µ0; Rd) contains a unique element of minimalnorm.

Theorem 4.8.7. Let λ 6 0, and let A be a λ-MMGR operator. Then:

(1) For each µ0 ∈ D(A) the curve s 7→ µs := Ssµ0 defined by (4.8.14) is theunique solution of the abstract Cauchy problem associated to A posed inDefinition 4.4.1, on [0,∞).

(2) The mapping ([0,+∞)×D(A) 3 (s, µ0)→ Ssµ0 ∈ D(A) can be extended ina unique way to a semigroup of Lipschitz operators

([0,+∞)×D(A) 3 (s, µ0)→ Ssµ0 ∈ D(A), (4.8.63)

166

Section 4.8

whose paths are continuous. Furthermore, we have the following estimates

W2(Stµ10, Ssµ

20) 6 e−λtW2(µ1

0, µ20) ∀ t > 0, ∀µ1

0, µ20 ∈ D(A), (4.8.64)

W2(St1µ0, St2µ0) 6 e−λmaxt1,t2|Aµ0|, ∀ t1, t2 > 0, ∀µ0 ∈ D(A).(4.8.65)

Furthermore, for each µ0 ∈ D(A), the semigroup path 0 6 t 7→ µt := Stµ0 has thefollowing properties:

(3) Any version (vt)t of the tangent Wasserstein velocity field of t 7→ µt satisfies

|vt|L2(µt,Rd) = |Aµt|, L1−a.e. t ∈ [0,+∞). (4.8.66)

Consequently, denoting ξt to be the unique vector in A(µt) of minimal norm,for t > 0, the vector field (ξt)t is Lebesgue measurable.

(4) The L1-a.e. everywhere defined metric derivative |µ|(t) of s 7→ µs at t has aright continuous version, the function 0 6 t 7→ e−λt|µ|(t) is non-increasing,t 7→ µt is right metrically differentiable at each t ∈ [0,+∞), and we havethat

limh↓0

W2(µt, µt+h)h

= |Aµt| ∀ t ∈ [0,+∞). (4.8.67)

Proof. Proving the claim in item (1) requires more effort than proving the remain-ing claims, and we will prove this claim first. To this aim, fix µ0 ∈ D(A), andt > 0, and let µs := Ssµ0, for s > 0. Due to (4.8.15), the curve 0 6 s 7→ µscontinuous, and of class AC2((0, t);P2(Rd), for each t > 0. Therefore, [5] Theorem8.3.1 and [5] Lemma 8.1.2 yield existence of a Borel field (vs)s>0 such that (4.4.5)and (4.4.6) hold, using a gluing argument. We need to show the existence of avector field (vs)s>0 which moreover satisfies (4.4.7).

For s ∈ [0, t], and n ∈ N, recall the numbers kns , and δns , defined in (4.8.44)and the measures µn,k defined (4.8.46). For each n ∈ N and s ∈ [0, t], define thefollowing 3-plans

γns :=(i,(−ξnkns+1

),

(1− δns

2−nt


δns2−nt

i

)#

µn,kns+1 (4.8.68)

and define moreover

dγn :=1t

dγns ds ∈ P([0, t]× R3d), for n ∈ N. (4.8.69)

It is not hard to see that the measurability conditions which are required in orderfor the integral at the right hand side in (4.8.69) to be well defined, are indeed

167


space P2(Rd)

satisfied. Moreover, for each n ∈ N, we have that∫[0,t]×R3d

(s2 + |x|2 + |y|2 + |z|2) dγn(s, x, y, z) 6t2

3+

2n−1∑k=0

2−nW 22 (µn,k+1, δ0)

+2n−1∑k=0

2−n‖ξnk+1‖2L2(µn,k+1;Rd) +1t

∫ t

0

W 22 (µns , δ0) ds,

(4.8.70)and since each curve s 7→ µns starts at µ0, (4.8.48) imply that

supn∈N

∫[0,t]×R3d

|u|2 dγn(u) < +∞. (4.8.71)

Notice that in particular the images of all curves [0, t] 3 s 7→ µns are contained insome ball in P2(Rd). Well, (4.8.71) implies that the 1-moments of the sequence(γn)n are uniformly integrable, hence there is a subsequence (γnr )r and a Borelmeasure γ ∈ P1([0, t]× R3d), such that

γnrr→+∞−→ γ, in P1([0, t]× R3d) and in P([0, t]× Rd). (4.8.72)

Moreover, by [5] Lemma 5.1.7 (consider γ to be a measure on R4d), we have thatγ ∈ P2([0, t]×R3d). Clearly (π0

4)#γ = 1tL

1[0,t] holds, since (π0

4)#γn = 1

tL1[0,t] holds,

for all n ∈ N. By Lemma 4.8.5, (4.8.72) implies that

γnrsr→+∞−→ γs, in P(R3d), L1–a.e. s ∈ [0, t], (4.8.73)

where dγs ds denotes the disintegration of γ, with respect to 1tL

1|[0,t]. A similarcomputation as in (4.8.70), together with a reasoning given in the lines below(4.8.70) gives a uniform bound in n ∈ N of the 2-moments of the sequence (γns )n,and that for each s ∈ [0, t] (and even uniformly). Therefore, for each s ∈ [0, t]such that the convergence in (4.8.73) holds, there is a subsequence of (γnrs )r,which converges in P1(R3d). However, due to (4.8.73), for each s ∈ [0, t] the onlypossible limit is γs, and we conclude that

γnrsr→+∞−→ γs in P1(R3d), L1–a.e. s ∈ [0, t]. (4.8.74)

Letγ := (π0,1,2

4 )#γ, (4.8.75)

a probability measure in P([0, t] × R2d). By Lemma 4.8.6, µns → µs in P2(Rd)holds. Therefore (4.8.72) implies that

dπ0,1# γ = dµs ds. (4.8.76)

Moreover by (4.8.74), we have that

νnrs := π1,2# γnrs

r→+∞−→ π1,2# γs =: νs, in P1(R2d), L1–a.e. s ∈ [0, t], (4.8.77)

168

Section 4.8

while for each n ∈ N and s ∈ [0, t],

νns =(i, ξnkns+1

)#µn,kns+1 (4.8.78)

holds by definition (see (4.8.69) and (4.8.68)). An easy application of Lemma 4.8.6gives that for each s ∈ [0, t], µnr,knrs +1

W2−→ µs, as r → +∞, so the first marginal ofthe measure νs (defined in (4.8.77)) equals µs, for L1-a.e. s ∈ [0, t]. Let us denotethe disintegration of νs with respect to µs by

dνs(x, y) = dνs,x(y) dµs(x), (for L1–a.e. s ∈ [0, t]). (4.8.79)

In light of [5] Theorem 5.4.4.(ii), (4.8.77), (4.8.79), and (4.8.78), we can concludethat

L2(µnrknrs +1

; Rd) 3 −ξnrknrs +1

−→ −ξs ∈ L2(µs; Rd) weakly as r → +∞,

− ξs(x) :=∫

Rdy dνs,x(y), x ∈ Rd, for L1–a.e. s ∈ [0, t]

(4.8.80)(−ξs is the barycenter of the disintegration in (4.8.79)). Now, since A is assumedto be a λ-MMGR operator, while for each n ∈ N and s ∈ [0, t] we have thatξnkns+1 ∈ A(µn(kns+1)t2−n) (see (4.8.43)) together with (4.8.80), item 2 of Definition4.4.10 guarantees that

ξs ∈ A(µs), L1–a.e. s ∈ [0, t]. (4.8.81)

The vector field [0, t]× Rd 3 (s, x) 7→ ξs(x) is clearly Borel, by the construction.Next, we are going to verify that the continuity equation for s 7→ µs (see

(4.8.54)) holds, with (vs)s := (−ξs)s. To this aim, pick a ψ ∈ C∞c ([0, t)×Rd), andwrite ψs := ψ(s, .), for s ∈ [0, t]. The convergence∫ t

0

∫Rd∂sψs dµnrs ds→

∫ t

0

∫Rd∂sψs dµs ds, as r →∞ (4.8.82)

holds due to the bounded convergence theorem, as the convergence∫

Rd ∂sψs dµnrsn→+∞−→

∫Rd ∂sψs dµs holds for each s ∈ [0, t], and these functions are uniformly

bounded in s. On the other hand, due to the continuity equation for s 7→ µnrs onP2(Rd), we have that∫ t

0

∫Rd∂sψs dµnrs ds = −

∫Rdψ0 dµnr0 −

∫ t

0

∫Rd〈∇xψs, vnrs 〉dµnrs ds

= −∫

Rdψ0 dµ0 −

∫[0,t]×R3d

〈∇zψs(z), y〉dγnr (s, x, y, z)

−→ −∫

Rdψ0 dµ0 −

∫[0,t]×R3d

〈∇zψs(z), y〉dγ(s, x, y, z), as r → +∞.

(4.8.83)

169


space P2(Rd)

The convergence in (4.8.83) is justified by (4.8.72), and an application of [5] Lemma5.1.7, and [5] Proposition 7.1.5 (as |〈∇ψ(x), y〉| 6 c|y|, c := sup |∇ψ| < +∞). Sinceµnkns , µ

nkns+1

W2−→ µs as n→ +∞ holds for each s ∈ [0, t], [5] Proposition 7.1.3 yields

π0,2# γs

nr → (i, i)#µs as r → +∞, ∀s ∈ [0, t], (4.8.84)

which amounts to x = z γs-a.e., for L1-a.e. s ∈ [0, t]. Therefore,∫[0,t]×R3d

〈∇ψs(z), y〉dγ(s, x, y, z) =∫ t

0

∫R3d〈∇ψs(x), y〉dγ(s, x, y, z)

=∫ t

0

∫R2d〈∇ψs(x), y〉dνs ds =

∫ t

0

∫Rd

∫Rd〈∇ψs(x), y〉dνs,x(y) dµs(x) ds

=∫ t

0

∫Rd〈∇ψs(x),

∫Rdy dνs,x(y)〉dµs ds =

∫ t

0

∫Rd〈∇ψs(x),−ξs(x)〉dµs ds.

(4.8.85)Finally, (4.8.82), (4.8.83), and (4.8.85), yield that the vector field (−ξs)s is a ver-sion of a Wasserstein-2 velocity vector field of the curve [0, t] 3 s 7→ µs. Since t > 0was arbitrary, while (4.8.81) holds, a simple gluing argument completes the proofof the fact that 0 6 s 7→ µs is a solution of the abstract Cauchy problem associatedto A. Since we already proved uniqueness of such solutions in Proposition 4.8.1,item (1) of the theorem has been proven.

Next, let us prove the claims in item (2). (4.8.64), and (4.8.65) are alreadyproven in Theorem 4.8.1, and since for each t > 0, the mapping St : D(A)→ D(A)is Lipschitz, this mapping can be extended to a Lipschitz mapping St : D(A) →D(A), and the extension is unique. Denote the extension also by St, for each t > 0.Due to the uniqueness proven in Theorem 4.8.1, we have that S : [0,+∞)×D(A)→D(A) is a semigroup, i.e. for s, t > 0, Ss St = Ss+t holds. To prove that theextension S : [0,+∞) × D(A) → D(A) is also a semigroup, choose µ0 ∈ D(A),and let ε > 0. Then, there is a ν0 ∈ D(A), such that W2(µ0, ν0) < ε, and in lightof (4.8.64), we estimate for s, t > 0,

W2(Ss+tµ0, SsStµ0) 6 W2(Ss+tµ0, Ss+tν0) +W2(Ss+tν0, SsStν0)

+W2(SsStν0, SsStµ0) 6 e−λ(s+t)W2(µ0, ν0) + 0 + e−λse−λtW2(ν0, µ0)

62e−λ(s+t)ε.

Since ε > 0 was arbitrary, we conclude that Ss+tµ0 = SsStµ0. Thus (St)t>0

is a semigroup on D(A). The continuity of the paths 0 6 s 7→ Ssµ0, for eachµ0 ∈ D(A), follows easily by (4.8.64), and (4.8.65).

We proceed with proving the claims in item (4), and we first show that

limt↓0

W2(µ0, Stµ0)t

= |Aµ0| (4.8.86)

holds. Well, let t > 0, and choose h = t/m, n = 1, δ = t in (4.8.32), to obtain for

170

Section 4.8

each m ∈ N

W 22

(Jmtmµ0,Jtµ0

)6 |Aµ0|2 max

(1 +

λt

m

)−(m+2)

, (1 + λt)−3

·

[(t− t) + (m− 1)

λt

mt

]2

+(t

n+ t

)min t, t

.

(4.8.87)

Let m→∞ in (4.8.87), to obtain

W 22 (Stµ0,Jtµ0) 6 c|Aµ0|2t4 (4.8.88)

for a finite constant c > 0. In particular, we have that

lim supt↓0

W2(Stµ0,Jtµ0)t

= 0. (4.8.89)

Since µ0 ∈ D(A) is assumed, (4.7.43) and (4.8.89) imply that (4.8.86) holds,indeed. As the metric derivative |µ| is defined L1-a.e. in [0,+∞), it also coincideswith the left metric derivative |µ|+(t) := limh↓0

W2(µt+h,µt)h , for L1-a.e. t > 0,

hence (4.8.86) and the semigroup property of (St)t>0 imply that (4.8.67) holds,and also that s 7→ µs is right metrically differentiable at each t > 0. Moreover wehave that

|µ|(t) = |Aµt| L1−a.e. t ∈ [0,+∞). (4.8.90)

On the other hand, (4.8.65) implies that

|µ|(t) 6 e−λt|Aµ0|, (4.8.91)

for s > 0 and (4.8.90) and (4.8.91), imply that the function 0 6 t 7→ g(t) :=eλt|µ|(t) is L1-a.e. equal to a non-increasing function g. Then the metric derivativeis L1-a.e. equal to t 7→ e−λtg(t), which is a right continuous function. We havecompleted the proof of all claims in item (4).

In the final paragraph of this proof, we prove the claims in item (3) of thetheorem. Well, for a fixed t > 0, (4.8.80), (5.4.13) in [5], and (4.8.48), give that

‖ξs‖L2(µs;Rd) 6 lim infr→∞

∥∥∥ξnrknrs +1

∥∥∥L2(µnr,k

nrs +1;Rd)

6 |Aµ0|, (4.8.92)

for L1-a.e. s ∈ [0, t]. Then, (4.8.92), and the semigroup property of (St)t>0, implythat this velocity vector field satisfies

‖vs‖L2(µs;Rd) = ‖ξs‖L2(µs;Rd) 6 |Aµs|, L1-a.e. s ∈ [0, t]. (4.8.93)

Now (4.8.93), (4.8.90), and the L1-a.e. on [0, t] uniqueness of the velocity field ofminimal norm (see [5] Theorem 8.3.1), directly imply that

ξs ∈ TanµsP2(Rd), L1-a.e. s ∈ [0, t], (4.8.94)

and since the opposite inequality as in (4.8.93) holds by definition, and t > 0was arbitrary, (4.8.66) is proven. The second claim in item (3) is obvious andomitted.

171


space P2(Rd)

In light of Theorem 4.8.7 and Definition 4.4.9, it became apparent, that thevectors in the image of A, which are truly required, are the ones associated tothe images of the resolvents, together with all the vectors of the minimal norm.Since each of these vectors ξ ∈ A(µ), µ ∈ D(A), in fact lies in the tangent spaceTanµP2(Rd), we may as well redefine λ-MMGR operators as subsets of

P2(Rd)×⋃

µ∈P2(Rd)

TanµP2(Rd). (4.8.95)

From the geometric point of view, this definition would be even a more naturalone than Definition 4.4.5.

4.9 Towards an application

As far as applications of the abstract theory developed in the previous sectionsof this chapter are concerned, the author has considered the most natural PDEto be treated by means of λ-MMGR operators, to be the non-symmetric Fokker-Planck equation which we analyzed in Chapter III. We have shown there that theassociated semigroup on P2(Rd) has quite similar properties, as the semigroup ofsolutions of the Cauchy problem associated to a λ-MMGR operator (see Theorem4.8.7). However, it seems that the non-symmetric Fokker-Planck equation doesnot fall within the scope of the theory of this chapter. As a matter of fact, even ifone considers the transport semigroup associated to a maximal monotone mappingb : Rd → Rd (such b generates a contraction semigroup (St)t>0 on Rd, and thetransport semigroup on P2(Rd) is given by Stµ := (St)#µ, for example if b growsat most linearly), it seems very hard to verify the λ-MGR property, if b is not agradient. In this regard, it seems rather interesting to understand this problembetter, and more generally to investigate whether there is a reasonable counterpartof the non-linear Hille-Yosida theorem for contraction semigroups on Hilbert spaces(see [17] Theoreme 4.1). The non-linear version of the Hille-Yosida theorem statesthat to any contraction semigroup (St)t>0 defined on a closed convex subset of aHilbert space H, with continuous paths, there corresponds a maximal monotoneoperator A ⊂ H ×H such that

limt↓0

Stx− xt

= −Ax, ∀x ∈ D(A), (4.9.1)

and the semigroup (St)t>0 in (4.9.1) is defined by the exponential formula. It maybe concluded that there are more steps to take in order to understand what thecounterpart of the Hille-Yosida theorem in the context of the contraction semigrouson P2(Rd) should be.

Nevertheless, we can consider to study the infinite dimensional Heat equationon a separable Hilbert space, by means of our approach. Such an equation isassociated to the covariance operator Q of a Gaussian measure on a Hilbert space,or more generally, to the covariance operator of a Gaussian measure on Banach

172

Section 4.9

space. This equation has been studied by classical means of linear functionalanalysis by various authors, and the basic theory about this equation can be foundin [32]. The program that we intend to execute in this regard, i.e. with the aid ofthe ideas of this chapter is the following.

1. We have to extend our theory of λ-MMGR operators to cover situationswhere the domain of the considered operator is not contained in the subsetPr2 (Rd) ⊂ P2(Rd) of regular measures. The operators are then no longerregular, and instead of defining A as the set of elements [µ, ξ], where µ ∈P2(Rd), and ξ ∈ L2(µ; Rd), they should consist of measures µ ∈ P2(R2d).More precisely, we will say that µ ∈ A(µ), for µ ∈ P2(Rd), and we might wantto impose the restriction that (π2

2)#µ should be a member of the geometrictangent space to P2(Rd) at µ (we refer to [43] for basic properties of thegeometric tangent spaces at non-regular measures).

2. We have to extend our theory further to the Wasserstein-2-space associatedto the Cameron-Martin space associated to a Gaussian measure on a Hilbertspace, or possibly even to a Gaussian measure on a Banach space. TheMonge-Kantorovich problem and the Fokker-Planck equations in this con-text, have already been considered by various authors, see [41], [96], [40],and [70].

3. Once we have completed the two extensions in items 1 and 2, we intend toproceed by means of the finite dimensional projections. The finite dimen-sional projections of the infinite dimensional heat equation are easily seento give the gradient flow of the relative entropy functional on P2(Rd), butrelative to the rescaled inner product on Rn, which arises from the restric-tions of Q to finite dimensional subspaces of H. The main idea is to definethe operator (thus also its resolvents) as the limit of the finite dimensionalprojections.

If carried out successfully, our program will give new insights into the infi-nite dimensional equation—among other things, it will then be seen to induce acontraction semigroup with respect to the Wasserstein-2 distance induced by theCameron-martin space. In a finite dimensional setting (i.e. considering measureson Rd), the heat equation is the gradient flow of the entropy functional associ-ated to the Lebesgue measure. It is well known that there is no counterpart of theLebesgue measure in infinite dimensions, and this semigroup has no invariant statein finite nor in infinite dimension. Moreover, the semigroup on the linear space ofthe uniformly continuous functions of the solutions of the infinite dimensional heatequation., is not a differentiable semigroup, while the gradient flow semigroups dopossess an analogous property. These facts together may indicate that indeed theinfinite dimensional heat equation may not be a gradient flow equation, but per-haps a flow induced by an appropriately defined maximal monotone operator onthe Wasserstein-2 space corresponding to the associated Cameron-Martin space.The ideas presented in this section, are a work in progress by the author.

173

Chapter 5

Invariant measures forlocally Lipschitz stochasticdelay equations

This chapter has been accepted for publication as: I. Stojkovic and O. van Gaans,“Invariant measures and a stability theorem for locally Lipschitz stochastic delayequations” in Annales de l’Institut Henri Poincare (B) Probabilites et Statistiques.

We consider a stochastic delay differential equation with exponentially stable driftand diffusion driven by a general Levy process. The diffusion coefficient is assumedto be locally Lipschitz and bounded. Under a mild condition on the large jumpsof the Levy process, we show existence of an invariant measure. Main tools in ourproof are a variation-of-constants formula and a stability theorem in our context,which are of independent interest.

5.1 Introduction

The main purpose of this paper is to show existence of an invariant measure for astochastic delay differential equation of the form

dX(t) =

(∫[−α,0]

X(t+ s)µ(ds)

)dt+ F (X)(t−)dL(t), (5.1.1)

where L is a Levy process, α a positive real, and µ is a signed Borel measureon [−α, 0]. The diffusion coefficient F may be a function on R or a functionaldepending on the segment (X(t + s) : − α ≤ s ≤ t) of the solution X. If theunderlying deterministic equation, that is equation (5.1.1) with F = 0, is expo-nentially stable, it may be expected that the stochastic equation has an invariant

175

CHAPTER 5: Invariant measures and a stability theorem for locally Lipschitz stochastic

delay equations

measure under suitable conditions on F . Existence of invariant measures has beenshown for an increasingly general class of coefficients F . In 1982 Wolfe [108] dealtwith the case where µ is a (negative) point mass at 0 and F is a constant. In 2000,Gushchin and Kuchler [47] extended to the case of the general delay measure µ.More recently, Reiß et al. [92] considered nonlinear coefficients F . They assumea global Lipschitz condition on F , boundedness of F , and continuity of F withrespect to the Skorohod topology. In each of these works the analysis depends on avariation-of-constants formula for equation (5.1.1). In the case of global LipschitzF , such a formula has been proved in [93].

The theory of stochastic equations in a setting beyond globally Lipschitz con-ditions has been vastly extended during the last years in the field of stochasticpartial differential equations, see, e.g., [24, 28, 49, 78, 84, 38] for some recent de-velopments. Also in the field of stochastic delay differential equations there isinterest in results on equations with coefficients that are not globally Lipschitz. Inparticular, models in financial mathematics can naturally involve a combinationof delay, processes with jumps, and locally Lipschitz coefficients [97, (3.6)].

Our main contribution here is extending the results of [92] to equations withdiffusion coefficients that are only locally Lipschitz instead of globally Lipschitz.Moreover, the continuity with respect to the Skorohod topology is relaxed to acondition that is considerably better suited for verification in examples. Includedin our analysis is the proof of a variation-of-constants formula for (5.1.1) for locallyLipschitz F .

If the diffusion coefficient F is only locally Lipschitz with linear growth (seeDefinition 5.4.1 for the precise formulation), the eventual Feller property and thevariation-of-constants formula, which play a key role in [92, 93], do not followfrom the results given there. Establishing these results is the main content of thisarticle. We do so by approximating the locally Lipschitz diffusion coefficient byglobally Lipschitz coefficients in a suitable sense. The difficulty is to verify thatthe solutions of the equations with the approximated coefficients converge to thesolution of (5.1.1) in an appropriate sense and that the limit inherits the desiredproperties. It turns out that the reduction steps in the proof of the variation-of-constants formula in [93] have to be changed. The extension to locally Lipschitzcoefficients has to be done before increasing the generality of the other componentsand these steps have to be adapted accordingly. The proof of the eventual Fellerproperty is based on new estimates, which relax the conditions on F even in theglobally Lipschitz case. Moreover, we prove a stability theorem.

Two comments on the form of (5.1.1) are in order. First, in the spirit of [87,Chapter V] we present our results for the one dimensional equation. At the costof more complicated notation our arguments can be extended to equations in Rn.Second, (5.1.1) is formulated with a linear drift term. However, nonlinear driftterms are covered as well by our theory, due to the generality of the noise processesthat we allow. By doubling the dimension and including deterministic componentsin the process L, locally Lipschitz nonlinearities in the drift are included.

One may wonder how rich is the scope of equations with bounded locally Lips-chitz coefficients. On one hand, our generalization is interesting from a theoretical

176

Section 5.2

point of view. An important question is what stochastic perturbations that can beadded to a stable linear delay differential equation so that the stability persists inthe form of existence of an invariant measure. Our results are a natural step in ex-panding the generality if this theory. On the other hand, natural transformationsof equations with unbounded globally Lipschitz coefficients may yield equationswith bounded coefficients that are only locally Lipschitz, a situation that fits inour setting (see Example 5.7.4).

For other approaches to stochastic delay differential equations and invariantmeasures, see, e.g., [71, 74, 75].

The outline of our arguments is roughly as follows. If the diffusion coefficientF in (5.1.1) maps the Skorohod space of real valued cadlag functions on [−α, 0]into itself and satisfies a suitable locally Lipschitz and growth condition, then itis known that equation (5.1.1) has a unique solution X for any initial process on[−α, 0] (see [52]). The solution itself is, however, not a Markov process. Instead,one can consider the segments Xt = (X(t+a))−α≤a≤0 of the solution process. If asolution X(t) of equation (5.1.1) is such that all of the segments Xt have the samedistribution, then the solution itself is stationary as well. Therefore we want toapply the Krylov-Bogoliubov method to the segment process, and for that we needthe state space to be separable. If the driving process L has continuous paths,(Xt)t takes values in C[−α, 0], which is separable with the supremum norm ‖.‖∞.In general, L may have jumps and then (Xt)t is a process with as state space theSkorohod space D[−α, 0] of cadlag functions. This space is not separable under‖.‖∞, but it is separable when endowed with the Skorohod metric. In order toapply the Krylov-Bogoliubov method and obtain an invariant measure, we need theeventual Feller property (see (5.4.5) and (5.4.6) in Section 5.4 below) and tightnessof the segment process given by (5.1.1). We follow the approach of [92], wherethe tightness is obtained by means of suitable estimates on the semimartingalecharacteristics.

In Section 5.2 we give a brief review of the facts about the Skorohod space anddeterministic delay equations that we need in the sequel. In Section 5.3 we obtainthe variation-of-constants formula. In Section 5.4 we give a precise formulationof the input of equation (5.1.1) and introduce the segment process. Section 5.5deals with tightness of the segment process. The stability theorem is proved inSection 5.6 and the Markov and eventual Feller properties and the existence of aninvariant measure are established in Section 5.7.

5.2 Preliminaries

All the processes we consider are defined on the same filtered probability space(Ω,Ft,F ,P). Since we are going to work with a Markov process whose state spaceis the Skorohod space we recall some facts about it. For a < b, let D[a, b] andD[a,∞) denote the linear spaces of all real-valued cadlag functions defined on[a, b] and [a,∞), respectively. Similarly, for t0 > 0, let D[0, t0] denote the space ofadapted cadlag processes on [0, t0] and likewise D[0,∞). On D[a,∞) the Skorohod

177


delay equations

metric is given by

dS(ϕ,ψ) = infλ∈Λ[a,∞)

(‖ϕ λ− ψ‖∞ + |||λ|||) ,

where Λ[a,∞) is the set of all increasing [a,∞)→ [a,∞) homeomorphisms and

|||λ||| = supa≤s<t, s 6=t

∣∣∣∣logλ(t)− λ(s)

t− s

∣∣∣∣ .The space (D[a,∞), dS) is complete and separable. Similarly, there is a completeseparable metric on D[a, b], which we also denote by dS .

We will also use a weaker metric on D[−α, 0]. Consider an arbitrary β > α.We extend a ϕ ∈ D[−α, 0] to ϕ ∈ D[−β, 0] by ϕ(t) = ϕ(t) for t ∈ [−α, 0] andϕ(t) = 0 for t ∈ [−β,−α). Let the metric dβ on D[−α, 0] be defined by

dβ(ϕ,ψ) := dD[−β,0](ϕ,ψ), ϕ, ψ ∈ D[−α, 0],

where dD[−β,0] denotes the Skorohod metric on D[−β, 0]. It is straightforward toverify that dβ(ϕ,ψ) ≤ dS(ϕ,ψ) for all ϕ,ψ ∈ D[−α, 0], where dS still denotes theSkorohod metric on D[−α, 0]. The metric dβ is actually independent of β and onecould even choose β =∞.

Lemma 5.2.1. 1. (D[−α, 0], dβ) is a Polish space.

2. dβ and dS generate the same Borel σ-algebra B(D[−α, 0]).

Proof. 1. The set A := ϕ : ϕ ∈ D[−α, 0] is a closed subset of D[−β, 0],which is a Polish space. Indeed, Skorokhod convergence implies almost every-where convergence and we can finish the argument by right continuity of thelimit. 2. As dβ ≤ dS , the dβ-Borel σ-algebra is contained in the dS-Borel σ-algebra. For the opposite inclusion it is enough to show that finite dimensionalsets, that is, ϕ ∈ D[−α, 0] : (ϕ(s1), . . . , ϕ(sn)) ∈ C, where C ∈ B(Rn) and−α ≤ s1 ≤ · · · ≤ sn ≤ 0, are in the dβ-Borel σ-algebra (see [11, (15.2) on p. 157]).This is obvious as D[−α, 0], dβ is a subspace of (D[−β, 0], dD[−β,0])

Next we collect some results on the deterministic delay equation

x(t) = ϕ(0) +∫ t

0

(∫[−α,0]

x(s+ a)µ(da))

ds for t ≥ 0,x(a) = ϕ(a) for a ∈ [−α, 0].

(5.2.1)

Here α > 0, µ is a finite signed Borel measure on [−α, 0], and the initial conditionϕ ∈ D[−α, 0]. The results that we need can be found in a more general frameworkin [35]. However, we can give these results in a more easily accessible way asfollows. According to [33, Theorem (i), p. 972], for each ϕ ∈ D[−α, 0], there existsa unique function x : [−α,∞) → R whose restriction to [0,∞) is continuous andwhich satisfies (5.2.1). Indeed, the map ψ 7→

∫[−α,0]

ψ(a)µ(da) is a bounded linearmap from C[−α, 0] with the uniform norm into R. Therefore it is also bounded on

178

Section 5.3

the Sobolev space W 1,1[−α, 0], as W 1,1 is continuously embedded in Cb. Since eachϕ ∈ D[−α, 0] is bounded and measurable, we have (ϕ(0), ϕ) ∈M1 := R×L1[−α, 0].Hence we may apply [33, Theorem (i)] to obtain a unique solution x to (2.6) of[33]. By Fubini theorem x is the unique solution of (5.2.1).

To stress the dependence on the initial condition, we denote the solution of(5.2.1) by x(·, ϕ). The solution corresponding to initial condition ϕ(s) = 0 for−α ≤ s < 0 and ϕ(0) = 1 is called the fundamental solution and denoted by r.

The following variation-of-constants formula for x(·, ϕ) holds,

x(t, ϕ) = r(t)ϕ(0) +∫ t

0

r(t− s)∫

[−α,−s)ϕ(s+ a)µ(da)ds, t ≥ 0, (5.2.2)

where the inner integral is considered to vanish for s > α, hence the outer integralis actually from 0 to t ∧ α. Indeed, by Fubini and substitutions one can verifythat the right hand side of (5.2.2) satisfies (5.2.1) and therefore equals the uniquesolution x(·, ϕ). By similar arguments one can rewrite formula (5.2.2) as

x(t, ϕ) = ϕ(0)r(t) +∫

[−α,0]

∫ 0

s

r(t+ s− a)ϕ(a)daµ(ds) for t ≥ 0. (5.2.3)

The delay equation (5.2.1) is said to be stable if the fundamental solution rconverges to zero as t→∞. The condition

v0(µ) := sup

Reλ : λ ∈ C, λ−

∫[−α,0]

eλsµ(ds) = 0

< 0 (5.2.4)

implies the even stronger property of exponential stability of all solutions , i.e.,there exist γ,K > 0 such that |x(t, ϕ)| ≤ Ke−γt‖ϕ‖∞ for all t ≥ 0 and for anysolution x(·, ϕ) of (5.2.1). Indeed, for the stability of the fundamental solutionsee the text below Corollary 4.1 on p. 182 of [48], and then the exponentrialstability of arbitrary solutions with initial condition ϕ ∈ D[−α, 0] follows by directcomputation from (5.2.2).

It is clear from (5.2.1) that each of its solutions x(·, ϕ) is absolutely continuouson [0, T ] for every T > 0 and even continuously differentiable on (α,∞). If (5.2.4)holds, then (5.2.1) yields the exponential decay of the derivative x(·, ϕ) directly.

5.3 Variation-of-constants formula

This section establishes a variation-of-constants formula for equation (5.1.1). Themajor point is to show existence and uniqueness for equations of variation-of-constants form.

Recall that for two local martingales M and N their quadratic covariationprocess is denoted by [M,N ]. Recall also that for t ≥ 0,

∫ t0|dA(s)| (

∫∞0|dA(s)|)

denotes the pathwise total variation on [0, t] ([0,∞), respectively) of a processA ∈ D[0,∞). The next two definitions are taken from [87, Section V.2, p. 250–251].

179


delay equations

Definition 5.3.1. Let 1 ≤ p ≤ ∞. For a process X ∈ D[0,∞) set

‖X‖Sp := ‖ supt≥0|X(t)| ‖Lp

and let Sp[0,∞) denote the Banach space of X ∈ D[0,∞) for which ‖X‖uuSp isfinite. Further for a local martingale M with M(0) = 0 and a cadlag adaptedprocess A with paths of finite variation on compact intervals a.s. and A(0) = 0,set

jp(M,A) := ‖[N,N ]1/2∞ +∫ ∞

0

|dA(s)|‖Lp .

For a semimartingale Z with Z(0) = 0 set

‖Z‖Hp := infZ=M+A

jp(M,A),

where the infimum is taken over all possible decompositions Z = M + A whereM is a local martingale with M(0) = 0 and a A is a cadlag adapted proces withpaths of finite variation on compact intervals a.s. and A(0) = 0. Furthermore, letHp[0,∞) denote the Banach space of all semimartingales Z with Z(0) = 0 suchthat ‖Z‖Hp is finite. For t0 > 0, the analogous Banach spaces of processes onlydefined on [0, t0] are denoted by Sp[0, t0] and Hp[0, t0].

For 1 ≤ p <∞ there exists a constant cp such that

‖Z‖Sp ≤ cp‖Z‖Hp (5.3.1)

for all semimartingales Z with Z(0) = 0 (see [87, Theorem V.2, p.252]. For aprocess X ∈ D[0,∞) and a stopping time T , the pre-stopped process XT− is givenby

(XT−)(t)(ω) = X(t)(ω)10≤t<T (ω)+X(t ∧ T (ω)−)(ω)1t≥T (ω)>0, ω ∈ Ω, t ≥ 0.

Observe that XT− ∈ D[0,∞).The following definition from [93] is essentially from [87, p. 256].

Definition 5.3.2. A map Ψ: D[0,∞)→ D[0,∞) is called functional Lipschitz iffor any X,Y ∈ D[0, t0]

(a) for any stopping time T , XT− = Y T− implies Ψ(X)T− = Ψ(Y )T−

(b) there exists a (positive finite) adapted increasing process K such that

|Ψ(X)(ω, t)−Ψ(Y )(ω, t)| ≤ K(ω, t) sups≤t|X(ω, s)− Y (ω, s)|.

Definition 5.3.3. A map Ψ: D[0,∞)→ D[0,∞) is called locally Lipschitz func-tional with linear growth if it satisfies

180

Section 5.3

(a) for all X,Y ∈ D[0,∞) and all stopping times T

XT− = Y T− =⇒ Ψ(X)T− = Ψ(Y )T−

(b) for each n ∈ N there exists an adapted increasing process Kn such that forall t ≥ 0 and all ω ∈ Ω,

|Ψ(X)(ω, t)−Ψ(Y )(ω, t)| ≤ Kn(ω, t) sups≤t|X(ω, s)− Y (ω, s)|

whenever |X(ω, s)|, |Y (ω, s)| ≤ n for all s ≤ t

(c) there exists a positive increasing adapted process γ(t) such that

|Ψ(X)(ω, t)| ≤ γ(ω, t)(1 + sups≤t|X(ω, s)|) for all X ∈ D[0,∞).

Notice that by (b): Xt0 = Y t0 implies Ψ(X)t0 = Ψ(X)t0 , for any X,Y ∈D[0,∞). Therefore, for U ∈ D[0, t0] we can define Ψ(U) ∈ D[0, t0] unambiguously,simply by extending U to [0,∞). Notice also that any functional Lipschitz map isa locally Lipschitz functional with linear growth.

In the sequel we will need the following condition on a function g : [0, t0]→ R.

Condition 1 1. For every Y ∈ Sp[0, t0] and Z ∈ H∞[0, t0],∫ ·0

g(· − s)Y (s−)dZ(s) ∈ Hp[0, t0] and∥∥∥∥∫ ·0

g(· − s)Y (s−)dZ(s)∥∥∥∥Hp[0,t0]

≤ R‖Y ‖Sp[0,t0]‖Z‖H∞[0,t0],

where t0 > 0, R > 0 and 1 < p <∞ are given constants.

Lemma 5.3.4. Let t0 > 0 be given and let g satisfy Condition 1 for some p and R.Assume that Ψ is locally Lipschitz with linear growth, Ψ(0) = 0, and such that theprocesses Kn(t) and γ(t) are constants. Let Z ∈ H∞[0, t0] with ‖Z‖H∞ < 1

4cpRγ

and let J ∈ Sp[0, t0]. Then for any stopping time T the equation

X(t) = JT−(t) +(∫ ·

0

g(· − s)Ψ(X)(s−)dZ(s))T−

(t), 0 ≤ t ≤ t0

has a unique solution X ∈ Sp[0, t0]. Moreover, X is a semimartingale if J is.

Proof. For each n ∈ N and X ∈ D[0, t0] let X(n)(t) := (X(t)∧n)∨(−n), t ∈ [0, t0].Now define Ψn : D[0, t0]→ D[0, t0] by

Ψn(X) := Ψ(X(n)).

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delay equations

Then Ψn is functional Lipschitz: (a) is immediate and for (b) notice that

|Ψn(X)(t)−Ψn(Y )(t)| ≤ Kn sups≤t|X(n)(s)− Y (n)(s)|

≤ Kn sups≤t|X(s)− Y (s)|.

By [87, Theorem V.5(ii), p. 254], there is a stopping time Tn such that P(Tn >t0) > 1 − 2−n and ZT

n− ∈ S( 12cpKnR

) (notation of [87]), with cp as in (5.3.1).Then by [93, Lemma 5.6] there is a unique Xn ∈ Sp[0, t0] such that

Xn(t) = JT−(t) +(∫ t

0

g(t− s)Ψn(Xn)(s−)dZTn

(s))T−

and we have

‖Xn‖Sp ≤ ‖JT−‖Sp +

∥∥∥∥∥(∫ ·

0

g(· − s)Ψn(Xn)(s−)dZTn−(s)

)T−∥∥∥∥∥Sp

≤ 2‖J‖Sp + 2cp

∥∥∥∥∫ ·0

g(· − s)Ψn(Xn)(s−)dZTn−(s)

∥∥∥∥Hp

≤ 2‖J‖Sp + 2cpR‖Ψn(Xn)‖Sp‖ZTn−‖H∞

≤ 2‖J‖Sp + 2cpRγ(1 + ‖Xn‖Sp)2‖Z‖H∞ ,

where the second inequality follows from [87, Theorem V.2 and proof of TheoremV.5], the third by Condition 1, and the fourth by (c) of Definition 5.3.3. Since4cpRγ‖Z‖H∞ < 1 by assumption, we obtain

‖Xn‖Sp ≤2‖J‖Sp + 4cpRγ‖Z‖H∞

1− 4cpRγ‖Z‖H∞=: C <∞ for all n.

In particular

P( sup0≤s≤t0

|Xn(s)| > n) ≤ Cp

npfor all n ∈ N.

Let now An := sup0≤s≤t0 |Xn(s)| ≤ n and Bn := Tn > t0. ThenP(An∩Bn) ≥

1 − 2−n − Cp/np. Set Ωn := An ∩ Bn, Ωn :=⋂k≥n Ωk, Ω′ :=

⋃n Ωn, so that

Ωn ⊂ Ωn+1, P(Ωn) ↑ P(Ω′) = 1. On Ωn we have that

sup0≤s≤t0

|Xk(s)| ≤ k, ZTk− = Z,

and also Ψk1(Xk) = Ψk(Xk) = Ψ(Xk) for k1 ≥ k ≥ n.

182

Section 5.3

If we consider measures Pn given by Pn(A) := P(A∩Ωn)/P(Ωn) for n so largethat P(Ωn) > 0, we have for k ≥ n that

Xk(t) = JT−(t) +(∫ t

0

g(t− s)Ψk(Xk)(s−) dZ(s))T−

= JT−(t) +(∫ t

0

g(t− s)Ψ(Xk)(s−) dZ(s))T−

,

Xn(t) = JT−(t) +(∫ t

0

g(t− s)Ψn(Xn)(s−) dZ(s))T−

= JT−(t) +(∫ t

0

g(t− s)Ψ(Xn)(s−) dZ(s))T−

= JT−(t) +(∫ t

0

g(t− s)Ψk(Xn)(s−) dZ(s))T−

Pn-a.s., the stochastic integrals being computed according to the probability Pn.This is easily seen by applying [87, Theorem II.14 and Theorem II.18] and the factthat the stochastic convolutions above are P-a.s. cadlag. Since Ψk is functionalLipschitz in the sense of [93, Definition 5.1]), the uniqueness in [93, Proposition5.8] yields that we have for k ≥ n that Xk = Xn Pn-a.s., thus P-a.s. on Ωn. Hencethere is a process X such that for almost all ω ∈ Ω′ and all t ≥ 0

X(t, ω) = limn→∞

Xn(t, ω),

which is adapted and cadlag, since the filtration satisfies the usual conditions.Moreover, we have for each n that

X = Xn = JT− +(∫ ·

0

g(· − s)Ψn(Xn)(s−) dZTn−(s)

)T−= JT− +

(∫ ·0

g(· − s)Ψ(X)(s−) dZ(s))T−

on Ωn a.s. Hence X satisfies the equation. We also have that

sup0≤s≤t0

|X(s)|p = limn→∞

sup0≤s≤t0

|Xn(s)|p P-a.s.,

so that by Fatou’s lemma,

‖X‖Sp ≤ supn‖Xn‖Sp <∞.

For uniqueness, suppose X ′ is another solution in D[0, t0]. Consider the funda-mental sequences Sn := inft : |X(t)| > n and Sn1 := inft : |X ′(t)| > n. Thenon the set Cn := Sn ∧ Sn1 > t0 the processes X and X ′ both solve the equation

183


delay equations

with Ψn, so by uniqueness in [93, Proposition 5.8] X = X ′ on Cn. Hence X = X ′

a.s.Finally, if J is a semimartingale, XSn− is a semimartingale for each n, by (a)

and (c) of Definition 5.3.3 together with Condition 1, hence X is a semimartingaleas well.

Lemma 5.3.5. Assume the situation of Lemma 5.3.4 with p > 2, but without thecondition ‖Z‖H∞[0,t0] <

14cpRγ

. Then for each stopping time T the equation

X(t) = JT−(t) +(∫ t

0

g(t− s)Ψ(X)(s−) dZ(s))T−

, 0 ≤ t ≤ t0

has a unique solution in D[0, t0]. If J is a semimartingale, then X is a semi-martingale as well.

Proof. Let Ψn be the functional Lipschitz maps as in the proof of Lemma 5.3.4.By [93, Proposition 5.8] for each n there is an Xn ∈ D[0, t0] such that

Xn(t) = J(t) +∫ t

0

g(t− s)Ψn(Xn)(s−) dZ(s),

which is a semimartingale if J is a semimartingale. (Indeed, before applying [93,Proposition 5.8], J and Z can be extended constantly after t0 to [0,∞) and thenthe solution can be restricted to [0, t0].) Stop Xn at T− to obtain

(Xn)T−(t) = JT−(t) +(∫ t

0

g(t− s)Ψn((Xn)T−)(s−) dZ(s))T−

and set Xn := (Xn)T−. As before, Condition 1 and [87, Theorem V.2] yield

‖Xn‖Sp[0,t0] ≤ 2‖J‖Sp[0,t0] + 2cpR‖Ψn(Xn)‖Sp[0,t0]‖Z‖H∞[0,t0].

By the linear growth of Ψ we have

|Ψn(Xn)(t)| = |Ψ((Xn ∧ n) ∨ (−n))(t)|≤ γ(1 + sup

0≤s≤t|(Xn(s) ∧ n) ∨ (−n)|) ≤ γ(1 + n)

and obtain‖Xn‖Sp[0,t0] ≤ 2‖J‖Sp + 2cpRγ(1 + n)‖Z‖H∞[0,t0].

Let An := sup0≤t≤t0 |Xn(t)| > n, n ∈ N. Then

P(An) ≤ 2n−p(‖J‖Sp[0,t0] + cpRγ(1 + n)‖Z‖H∞[0,t0])

by Chebyshev’s inequality. Let further Ωn :=⋂k≥n(Ω \ Ak). Notice that Ωn ⊂

Ωn+1 and P(Ωn) ↑ 1, since p > 2. Define probability measures Pn(A) := P(Ωn ∩A)/P(Ωn) for n ≥ N , where N is such that P(ΩN ) > 0. Arguing as in Lemma 5.3.4we obtain for k ≥ n that Xk = Xn on Ωn, so we can define a process X := limXn,which is then the unique solution of the equation. If J is a semimartingale thenso is X.

184

Section 5.3

Proposition 5.3.6. Let Z be a semimartingale and J ∈ D[0,∞). Let g be afunction on [0,∞) which satisfies Condition 1 for each t0 > 0 with some R(t0) > 0and some fixed p > 2. Further, suppose that Ψ is a locally Lipschitz functionalsuch that the processes Kn and γ are deterministic. Then the equation

X(t) = J(t) +∫ t

0

g(t− s)Ψ(X)(s−) dZ(s), t ≥ 0,

has a unique solution in D[0,∞), which is a semimartingale if J is.

Proof. Fix t0 > 0. First we show existence and uniqueness on [0, t0]. There isa fundamental sequence T ` of stopping times such that JT

`− ∈ S∞[0,∞) andZT

`− ∈ H∞[0,∞). By the linear growth of Ψ and Condition 1, the convolu-

tion process(∫ t

0g(t− s)Ψ(0)(s−) dZ(s)

)t≥0

is a semimartingale. Hence we may

assume that Ψ(0) = 0.By the previous lemma, for each ` there is an X` such that

X`(t) = JT`−(t) +

(∫ t

0

g(t− s)Ψ(X`)(s−) dZT`−(s)

)T `−for all t ∈ [0, t0].

Moreover,

XT `−`+k (t) = JT

`− +(∫ t

0

g(t− s)Ψ(XT `−`+k )(s−) dZT

`−(s))T `−

.

Define X := limX` and argue as in the two previous lemmas to show that X isthe unique solution on [0, t0]. Finally, we can use the same techniques to patchthe solutions together obtaining a unique global solution.

We can now show existence and uniqueness for the general equation of variation-of-constants form.

Theorem 5.3.7. Let Ψ be a locally Lipschitz functional with linear growth andlet g be a function on [0,∞) which satisfies Condition 1 for each t0 > 0 withsome R(t0) > 0 and for some fixed p > 2. Suppose Z is a semimartingale andJ ∈ D[0,∞). Then the equation

X(t) = J(t) +∫ t

0

g(t− s)Ψ(X)(s−) dZ(s) (5.3.2)

has a unique solution in D[0,∞), which is a semimartingale if J is.

Proof. Fix t0 > 0. For n, k ∈ N there are constants cn,k > 0, γk > 0 such that

P(Ωn,k) ≥ 1− 2−n−k, where Ωn,k := Kn(t0, ω) ≤ cn,k

P(Ωk) ≥ 1− 2−k, where Ωk := γ(t0, ω) ≤ γk.

185


delay equations

Set Ωk :=⋂n Ωn,k ∩ Ωk. Then P(Ωk) ≥ 1 − 2−k+1, and on Ωk, Kn(t0, ω) and

γ(t0, ω) are bounded functions. Now apply the last part of the proof of Proposition5.8 of [93], using existence and uniqueness from the previous proposition.

As we mentioned in the introduction, the fundamental solution r of the deter-ministic delay equation is absolutely continuous on compacts and its derivative isbounded on compacts. Hence due to [93, Lemma 4.2], r satisfies Condition 1 forany t0 > 0 and p ≥ 1 with R = 1 + (1 + cp)t0 sup0≤t≤t0 |r

′(t)|. So there is a uniquesolution of the variation-of-constants formula (5.3.2) with g = r. This solutionalso satisfies the stochastic delay differential equation

dX(t) =

(∫[−α,0]

X(t+ s)µ(ds)

)dt+ Ψ(X)(t−)dZ(t). (5.3.3)

This can be shown by applying the stochastic Fubini theorem. The result isactually Lemma 6.1 in [93]. Although the statement there presupposes Ψ to befunctional Lipschitz, the only property of the functional Ψ used in the proof isthat it is a map D[0,∞) → D[0,∞). As (5.3.3) has a unique solution (see [52]),it follows that this solution satisfies the variation-of-constants formula. Statedprecisely:

Theorem 5.3.8. Let µ be a finite signed Borel measure on [−α, 0] and let r bethe fundamental solution of the deterministic delay equation. Let Ψ be a locallyLipschitz functional and let Z and J be semimartingales. The unique solution Xof

X(t) = X(0) + J(t) +∫ t

0

∫[−s,0]

X(s+ a)µ(da)ds

+∫ t

0

Ψ(X)(s−) dZ(s), t ≥ 0,

satisfies

X(t) = r(t)X(0) +∫ t

0

r(t− s) dJ(s)

+∫ t

0

r(t− s)Ψ(X)(s−) dZ(s), t ≥ 0.

5.4 The equation and the segment process

In the remaining part of the paper we consider (5.1.1) and show that it has aninvariant measure under suitable conditions.

Our approach is to see the stochastic equation (5.1.1) as a perturbation of thedeterministic equation (5.2.1). If the deterministic part is stable it is plausible toexpect existence of an invariant measure under mild conditions on the diffusion

186

Section 5.4

part. Therefore we assume that (5.2.4) holds. For an analysis of the case where(5.2.4) does not hold, see, e.g., [10].

As was shown in [47, Theorem 3.1], even if F is constant, a necessary conditionfor the existence of an invariant measure on the jumps of the Levy process L isthat

∫|x|>1

log |x| ν(dx) < ∞, where ν denotes the Levy measure of L. As oursituation is even more general, we need this condition as well.

As mentioned in the introduction, the main point of this paper is to relaxthe global Lipschitz condition in [92, Assumption 4.1(c)] to a locally Lipschitzcondition. Our locally Lipschitz condition on F is the following.

Definition 5.4.1. A map F : D[−α,∞)→ D[−α,∞) is called a locally Lipschitzfunctional of deterministic type (in short lolidet) if it satisfies

1 ∀n ∈ N ∃Kn > 0 such that ∀x, y ∈ D[−α,∞) ∀t ≥ 0

sups∈[t−α,t]

|x(s)|∨|y(s)| ≤ n =⇒ |F (x)(t)−F (y)(t)| ≤ Kn sups∈[t−α,t]

|x(s)−y(s)|,

2 ∃γ > 0 such that ∀x ∈ D[−α,∞) ∀t ≥ 0

|F (x)(t)| ≤ γ

(1 + sup

s∈[t−α,t]|x(s)|

).

Definition 5.4.2. Let (Φ(s))s∈[−α,0] be an initial condition, that is, a processwith cadlag paths and such that Φ(s) is F0-measurable for all −α ≤ s ≤ 0. ForX ∈ D[0,∞) or X ∈ D[−α,∞) define X Φ ∈ D[−α,∞) by

X Φ(s) :=

Φ(s), −α ≤ s < 0,X(s), s ≥ 0.

Here we extend the filtration by setting Fs := F0 for s < 0. Define furtherΨΦ : D[0,∞)→ D[0,∞) by

ΨΦ(X)(t, ω) := F (X Φ(·, ω))(t).

By standard arguments, ΨΦ indeed maps into D[0,∞).

Set Ω0 = ∅ and Ωn := sup[−α,0] |Φ(s)| > n for n ≥ 1. Then Ωn ↑ Ω. DefineCn : Ω→ R and γ : Ω→ R by

Cn(ω) := Kn on Ωn, Cn(ω) := Kn+k on Ωn+k \ Ωn+k−1

γ(ω) := γ(1 + n) on Ωn \ Ωn−1.

Notice that Cn and γ are F0-measurable for all n. Moreover, ΨΦ satisfies

1. for t ≥ 0 and X,Y ∈ D[0,∞),

|ΨΦ(X)(t, ω)−ΨΦ(Y )(t, ω)| ≤ Cn(ω) sups∈[(t−α)+,t]

|X(s, ω)− Y (s, ω)|

if sups∈[(t−α)+,t] |X(s, ω)| ∨ |Y (s, ω)| ≤ n, and

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delay equations

2.

|ΨΦ(X)(t, ω)| ≤ γ(ω)

(1 + sup

[(t−α)+,α]

|X(t, ω)|

)for all X ∈ D[0,∞).

In other words, ΨΦ satisfies (b) and (c) of Definition 5.3.3, and (a) is obvious.Define

JΦ(t) :=∫ t

0

∫[−α,−s)

Φ(s+ a)µ(da)ds, t ≥ 0.

Then JΦ is an adapted (by Fubini arguments) cadlag process of finite variation.Moreover, r(t)Φ(0) is an adapted process of finite variation. Hence by [93, Theorem4.1], r(t)Φ(0) +

∫ t0r(t− s) dJ(s) is a semimartingale.

Let X be the unique solution (see [52, Theorem 4.5]) of

X(t) = Φ(0) +∫ t

0

∫[−α,0]

X Φ(s+ a)µ(da)ds+∫ t

0

F (X Φ)(s−) dL(s)

= Φ(0) + JΦ(t) +∫ t

0

∫[−s,0]

X(s+ a)µ(da)ds

+∫ t

0

ΨΦ(X)(s−) dL(s).

By Theorem 5.3.8, X is also a solution of

X(t) = r(t)Φ(0) +∫ t

0

r(t− s) dJ(s)

+∫ t

0

r(t− s)ΨΦ(X)(s−) dL(s).

Because of (5.2.2) Theorem 5.3.8 takes the following form in the current setting.

Theorem 5.4.3. Let F : D[−α,∞) → D[−α,∞) be lolidet. Then for X ∈D[0,∞) and Φ ∈ D[−α, 0] the following two statements are equivalent:

1. X is the unique solution of

X(t) = Φ(0)+∫ t

0

∫[−α,0)

X Φ(s+a)µ(da)ds+∫ t

0

F (X Φ)(s−) dZ(s), t ≥ 0,

2. X obeys the variation-of-constants formula

X(t) = x(t,Φ) +∫ t

0

r(t− s)F (X Φ)(s−) dZ(s), t ≥ 0.

188

Section 5.4

Recall that for a process X ∈ D[−α,∞) we denote by (Xt)t≥0 the segmentprocess, which takes values in D[−α, 0] for each t. More precisely, Xt(s) = X(t+s)for −α ≤ s ≤ 0. We wish to show that the segment process is Markov. For this tobe true F obviously has to be autonomous in the sense of the following definition.

Definition 5.4.4. A map F : D[−α,∞) → D[−α,∞) is autonomous if for allx ∈ D[−α,∞) and all s, t ≥ 0,

F (x(s+ ·))(t) = F (x)(s+ t).

Assume that F is autonomous. For u ≥ 0 and (Y (s))−α≤s≤0 cadlag and Fu-measurable, we consider the equation

X(t) = Y (0) +∫ t

0

∫[−α,0]

X(s+ a)µ(da) ds

+∫ t

0

F (X)(s−) dLu(s), t ≥ 0,

X(t) = Y (t), −α ≤ t < 0,

where Lu(t) = L(t + u) − L(u), t ≥ 0. The underlying filtration Gut is (the rightcontinuous version of) σ(L(s+u)−L(u) : 0 ≤ s ≤ t)∨Fu. Denote by (Xu

Y (t))t≥−αthe unique solution of this equation and let (Xu

Y,t)t≥0 denote the correspondingsegment process.

For any F0-adapted initial condition Φ and t ≥ 0 the process XΦ := X0Φ

satisfies

XΦ(u+ ·)(t)−XΦ(u) = XΦ(t+ u)−XΦ(u)

=∫ t+u

u

∫[−α,0]

XΦ(s+ a)µ(da) ds+∫ t+u

u

F (XΦ)(s−) dL(s)

=∫ t

0

∫[−α,0]

XΦ(u+ ·)(s+ a)µ(da) ds+∫ t

0

F (XΦ(u+ ·))(s−) dLu(s),

where the latter equality holds by the fact that F is autonomous. The processLu is a Levy process relative to the filtration Fu+· and Gut ⊂ Ft+u for all t ≥ 0,hence Xu

XΦ,uis also a solution of the equation relative to the filtration Fu+· (see

[87, Theorem II.16]). Hence

XuXΦ,u

(t) = XΦ(t+ u) (5.4.1)

for all t ≥ 0, due to the strong uniqueness of the equation.Under additional conditions we will show below that the segment (XΦ,t)t is a

Markov process, that is,

E[1B(XuXΦ,u,t)|Fu] = AB(XΦ,u), (5.4.2)

whereAB = E1B(Xu

ϕ,t), ϕ ∈ D[−α, 0],

189


delay equations

for every B ∈ B(D[−α, 0]). Notice that XΦ,t is F/B(D[−α, 0])-measurable, sincethe Borel σ-algebra generated by the Skorohod topology B(D[−α, 0]) equals theσ-algebra generated by the finite dimensional set [11, (15.2) on p. 157]. ByBb(D[−α, 0]) we denote the space of bounded Borel (relative to the Skorohodtopology) functions on D[−α, 0]. For 0 ≤ s ≤ t we define

Ps,tf(ϕ) := Ef(Xsϕ,t−s), ϕ ∈ D[−α, 0], f ∈ Bb(D[−α, 0]). (5.4.3)

We will show that Ps,t maps Bb(D[−α, 0]) into Bb(D[−α, 0]), that Pu,t = Pu,sPs,tfor 0 ≤ u ≤ s ≤ t, and that Ps,t = P0,t−s.

Then XΦ,t is a homogeneous Markov process and the operators

Pt := P0,t (5.4.4)

form a Markovian semigroup. We will also show that (Pt)t≥0 is eventually Fellerin the following sense:

1. for f ∈ Cb(D[−α, 0]), t ≥ α,

Ptf ∈ Cb(D[−α, 0]); (5.4.5)

2. for t ≥ α, f ∈ Cb(D[−α, 0]),

lims↓t

Psf(ϕ) = Ptf(ϕ) uniformly in ϕ. (5.4.6)

Since ∆Xϕ(t) = F (Xϕ)(t−)∆L(t), and L is stochastically continuous, we havethat Xϕ is stochastically continuous, hence by [92, Lemma 3.2], the segment pro-cess Xϕ,t is stochastically continuous as well. So by bounded convergence andsubsequence arguments, 2. follows.

The proof of 1. will be more involved and is given in Section 5.7.In the sequel we will use several assumptions on the input to our equation

(5.1.1), so we list them here.

Assumption 1 1. 1. v0(µ) < 0 (see (5.2.4)).

2. The Levy measure ν of L satisfies∫|x|>1

log |x|dν(x) <∞.

3. F is lolidet.

4. F is autonomous and bounded.

We remark that boundedness of F implies growth condition 2. of Definition5.4.1.

190

Section 5.5

5.5 Tightness of segments

In this section we obtain the tightness of the segment process under Assumption1. We start by showing that for each fixed T ≥ 0, and any uniformly boundedsequence Φn of initial conditions (i.e. supn sup−α≤s≤0 supω∈Ω |Φn(s, ω)| < ∞),that

supn

P(supt≤T|XΦn(t)| > K)→ 0 as K →∞ (5.5.1)

(recall that XΦ denotes the solution of (5.1.1) with initial condition Φ), by showingthe following stronger result.

Theorem 5.5.1. Let Φn be a uniformly bounded sequence of initial conditions.Let further Assumption 1 hold. Then the laws of (XΦn(t+ s) : s ∈ [0, α])t≥0, n∈Nform a tight set. Consequently,

supn

supt≥0

P( sup0≤u≤α

|XΦn(t+ u)| > K)→ 0 as K →∞. (5.5.2)

Notice that (5.5.2) implies (5.5.1). In order to prove Theorem 5.5.1, we needseveral lemmas.

Lemma 5.5.2. Suppose v0(µ) < 0. The fundamental solution r of the determin-istic delay equation is C1 on [α,∞) and its total variation TV[−α,∞)r is finite.

Proof. Since r is absolutely continuous on [0, T ] for every T > 0, we have that fors > t and sn → s that∫

[−α,0]

r(sn + a)µ(da) =: g(sn)→ g(s) :=∫

[−α,0]

r(s+ a)µ(da),

by bounded convergence. So h(t) :=∫ tαg(s) ds, t > α, is an antiderivative of a

continuous function, hence C1, and h′(t) = g(t), which is a continuous functionon [α,∞). Moreover, the estimate |g(t)| ≤ C ′ exp(−βt) holds for some C ′, β > 0,so that TV[α,∞)r ≤

∫∞αC ′ exp(−βt) dt < ∞. Since r is absolutely continuous on

[0, α] and TV[−α,0]r = 1, we obtain that TV[−α,∞)r <∞.

Lemma 5.5.3. Suppose v0(µ) < 0 and let r(t) :=∫

[−α,0]r(t+ a)µ(da), for t ≥ 0.

Then r is almost everywhere on [0,∞) equal to the derivative of r and the totalvariation TV[0,∞)r is finite.

Proof. The first claim follows directly. For the second claim, let 0 ≤ t0 < t1 <· · · < tn <∞, then

n−1∑i=1

|r(ti+1)− r(ti)| ≤∫

[−α,0]

n−1∑i=1

|r(ti+1 + a)− r(ti + a)| |µ|(da)

≤ TV[−α,∞)r|µ|.

191


delay equations

Lemma 5.5.4. Suppose v0(µ) < 0. Let A be a subset of D[−α, 0] such thatsupϕ∈A sup−α≤s≤0 |ϕ(s)| <∞. Then the solutions of the deterministic delay equa-tion satisfy

supϕ∈A

supt∈[−α,∞)

|x(t, ϕ)| <∞.

Proof. By (5.2.3), for ϕ ∈ A,

|x(t, ϕ)| =

∣∣∣∣∣ϕ(0)r(t) +∫

[−α,0]

∫ 0

s

r(t+ s− a)ϕ(s) dsµ(ds)

∣∣∣∣∣≤ supϕ∈A|ϕ(0)| sup

s∈[−α,∞)

|r(s)|+ sups∈[−α,∞)

|r(s)|α supϕ∈A

sup−α≤s≤0

|ϕ(s)||µ| <∞.

Lemma 5.5.5. If Assumption 1 holds, L(XΦn(t)) : t ≥ 0, n ∈ N is a tight setof laws on R, provided that supn sup[−α,0] supω∈Ω |Φn(s, ω)| <∞.

Proof. Since XΦn(t) = x(t,Φn) +∫ t

0r(t − s)F (XΦn)(s−) dL(s) and by Lemma

5.5.4 supt,n |x(t,Φn)| < ∞, we can execute the same proof as in [92, Proposition4.2], as the only property of F (XΦn) used there is that it is a bounded process.

We proceed by showing that the laws of the deducted segments XΦn(t + ·) −XΦn(t), t ≥ 0, are tight as well.

Define processes

In(u) :=∫

[−α,0]

XΦn(u+ v)µ(dv)

=∫

[−α,0]

(x(u+ v,Φn) +

∫ u+v

0

r(u+ v − s)F (XΦn)(s−) dL(s))µ(dv)

=∫

[−α,0]

x(u+ v,Φn)µ(dv) +∫ u

0

r(u− s)F (XΦn)(s−) dL(s),

where we used the stochastic Fubini theorem and that r(s) = 0 for s < 0. Hence,since XΦn is cadlag, the processes

V n(u) :=∫ u

0

r(u− s)F (XΦn)(s−) dL(s)

have cadlag versions, which we will use in the sequel.

Lemma 5.5.6. Processes V n defined above satisfy

limK→∞

supn∈N

supt≥0

P( sup0≤s≤α

|V n(t+ s)| > K) = 0,

in other words, sup0≤s≤α |V n(t+ s)| : t ≥ 0, n ∈ N is a tight set of laws on R.

192

Section 5.5

Proof. First we show that the set of laws L(V n(u)) : u ≥ 0, n ∈ N is tight. Todo so, we examine the proof of [92, Proposition 4.2]. There the authors prove thatthe family of laws L(X(t)) : t ≥ 0 is tight, where

X(t) =∫ t

0

r(t− s)F (X)(s−) dL(s), t ≥ 0.

In the proof they use the boundedness of F (X(s−)), the fact that r(t) decaysexponentially for t → ∞, and that the Levy process L is exactly as in our As-sumption 1. Due to our Assumption 1 we have the same bound for F (XΦn)(s−)for all n simultaneously. As we also have exponential decay of the function r, wecan execute the same proof for V n(u) and obtain that L(V n(u)) : u ≥ 0, n ∈ Nis tight.

Assup

0≤s≤α|V n(t+ s)| ≤ sup

0≤s≤α|V n(t+ s)− V n(t)|+ |V n(t)|,

and we have that L(V n(t)) : t ≥ 0, n ∈ N is tight, it is enough to show tightnessof the laws of sup0≤s≤α |V n(t+ s)− V n(t)|, where n ∈ N and t ≥ 0.

Our Levy process L decomposes into

L(t) = bt+M(t) +N(t),

where N(t) =∑s≤t ∆L(s)1|∆L(s)|>1, M is a square integrable Levy martingale,

and b ∈ R. Then

sup0≤u≤α

∣∣∣∣∫ t+u

0

r(t+ u− s)F (XΦn)(s−) dN(s)∣∣∣∣

≤∑s≤t+α

C ′m exp(−β(t− s))|∆N(s)|,

and the last process is bounded in probability by time reversal for compoundPoisson processes and [47, Lemma 4.3] (see also [92, Proof of Proposition 4.2]).Further, ∣∣∣∣∫ t+u

0

r(t+ u− s)bds∣∣∣∣ ≤ C ′b∫ t+u

0

exp(−β(t+ u− s)) ds < C ′b.

Therefore it is enough to show that the laws of

sup0≤u≤α

∣∣∣∣∫ t+u

0

r(t+ u− s)F (XΦn)(s−) dM(s)−∫ t

0

r(t− s)F (XΦn)(s−) dM(s)∣∣∣∣ ,

n ∈ N, t ≥ 0, are a tight family. Now∣∣∣∣∫ t+u

0

r(t+ u− s)F (XΦn)(s−) dM(s)−∫ t

0

r(t− s)F (XΦn)(s−) dM(s)∣∣∣∣

≤∣∣∣∣∫ t

0

(r(t+ u− s)− r(t− s))F (XΦn)(s−) dM(s)∣∣∣∣

+∣∣∣∣∫ t+u

t

r(t+ u− s)F (XΦn)(s−) dM(s)∣∣∣∣ .

193


delay equations

For the first term we estimate∣∣∣∣∫ t

0


=∣∣∣∣∫ t

0

∫ u

0

dr(t− s+ v)F (XΦn)(s−) dM(s)∣∣∣∣

=∣∣∣∣∫ t

0

∫ t+u

0

1t−v≤s≤t−v+uF (XΦn)(s−) dr(v) dM(s)∣∣∣∣

=∣∣∣∣∫ t+u

0

(∫ t−v+u

t−vF (XΦn)(s−) dM(s)

)dr(v)

∣∣∣∣≤∫ t+α

0

supw≤α, n∈N

∣∣∣∣∫ t−v+w

t−vF (XΦn(s−) dM(s)

∣∣∣∣ d|r|(v),

by the stochastic Fubini theorem. Hence

E supu≤α

∣∣∣∣∫ t

0


≤∫ t+α

0

(E supu≤α

∣∣∣∣∫ t−v+u

t−vF (XΦn)(s−) dM(s)

∣∣∣∣2)1/2

d|r|(v)

≤ (TV[0,∞)r)4m2(EM(α)2)1/2,

by Doob’s inequality, boundedness of F and the fact that M is a Levy squareintegrable martingale.

For the second term, we first extend r by r(s) = r(0) for s < 0 and compute∣∣∣∣∫ t+u

t

r(t+ u− s)F (XΦn)(s−) dM(s)∣∣∣∣

=∣∣∣∣∫ t+u

t

(∫ u

0

dr(t− s+ v) + r(t− s))F (XΦn)(s−) dM(s)

∣∣∣∣≤∣∣∣∣∫ t+u

t

∫ u

0

dr(t− s+ v)F (XΦn)(s−) dM(s)∣∣∣∣

+ |r(0)F (XΦn)(t−)∆M(t)|

≤∣∣∣∣∫ t+u

t

∫ u

0

1t−z≤s≤t+u−zF (XΦn(s−)) dr(z) dM(s)∣∣∣∣+ |r(0)|m

=∣∣∣∣∫ u

0

(∫ t+u−z

t

F (XΦn)(s−) dM(s))

dr(z)∣∣∣∣+ |r(0)|m,

by applying stochastic Fubini theorem and because |∆M | ≤ 1. Hence, arguing asabove, we obtain

E sup0≤u≤α

∣∣∣∣∫ t+u

t

r(t+ u− s)F (XΦn)(s−) dM(s)∣∣∣∣

≤ (TV[0,∞)r)4m2(EM(α)2)1/2 + |r(0)|m.

194

Section 5.6

Now the proof is complete.

Proposition 5.5.7. Let Assumption 1 hold, and let Φn be a uniformly boundedsequence of initial conditions. Then the laws of (XΦn(t+ s)−XΦn(t) : s ∈ [0, α]),t ≥ 0, n ∈ N, are tight in D[0, α].

Proof. We use the same strategy as in [92, Proposition 4.3] and only need to showthat

limK→∞

supn

supt≥0

P( sup0≤u≤α

|In(t+ u)| > K)→ 0,

where

In(u) =∫

[−α,0]

XΦn(u+ v)µ(dv)

=∫

[−α,0]

x(u+ v,Φn)µ(dv) +∫ u

0

r(u− s)F (XΦn)(s−) dL(s).

Since the first term is bounded in n and t by Lemma 5.5.4, we infer the claim withthe aid of Lemma 5.5.6.

Now proving Theorem 5.5.1 is straightforward.

Proof of Theorem 5.5.1. We haveXΦn(t+·) = (XΦn(t+·)−XΦn(t)1)+XΦn(t)1 forall t ≥ 0, where 1(s) = 1 for all s ∈ [0, α]. Let ε > 0. By Proposition 5.5.7, thereexists a compact set K ⊂ D[0, α] such that P(XΦn(t+ ·)−XΦn(t)1 ∈ K) ≥ 1−ε/2for all t ≥ 0 and n ∈ N. By Lemma 5.5.5 there exists a bounded interval I ⊂ Rsuch that P(XΦn(t) ∈ I) ≥ 1 − ε/2. Let K ′ := σ + c1 : σ ∈ K, c ∈ I. ThenP(XΦn(t+ ·) ∈ K ′) ≥ 1−ε. The set K ′ has compact closure in D[0, α], due to [11,Theorem 12.4]. Indeed, as K is compact it satisfies conditions (12.25) and (12.30)of [11]. Then K ′ satisfies these conditions as well, hence it has compact closure bythe same theorem. Thus, L(XΦn(t+ ·)) : t ≥ 0, n ∈ N is t ight.

5.6 A stability theorem

In this section we prove that Φn → Φ in D[−α, 0] w.r.t. dβ in probability impliesuniform convergence on compact sets in probability of the corresponding solutions,under Assumption 1 and the following condition:

ϕn → ϕ in D[−α, 0] w.r.t. dβ =⇒∫ α

0

(F (x ϕ)(t)− F (x ϕn)(t))2 dt→ 0,

(5.6.1)where x ϕ(t) := x(t) for t ≥ 0 and x ϕ(t) := ϕ(t) for −α ≤ t ≤ 0, and likewise forx ϕn .

We need the following approximations of F :

FN (x)(t) := F (x(N))(t), where xN := (x∧N)∨(−N), for x ∈ D[−α,∞), t ≥ 0, and N > 0.

195


delay equations

Then FN is Lipschitz in the sense of [92, (2.5)]. Indeed, for x, y ∈ D[−α,∞) andt ≥ 0 and N > 0, we have that |xN (t)|, |yN (t)| ≤ N for all t ≥ −α, hence bycondition 1 of Definition 5.4.1 there is KN such that

|FN (x)(t)− FN (y)(t)| = |F (xN )(t)− F (yN )(t)| ≤KN sup

s∈[t−α,t]|xN (s)− yN (s)| ≤ KN sup

s∈[t−α,t]|x(s)− y(s)|

for all t ≥ 0.If (5.6.1) holds for F then it also holds for the approximations FN , provided

N > supn,s |ϕn(s)| ∨ |ϕ(s)|, since then for any x ∈ D[0,∞),

FN (x ϕ) = F ((x ϕ)(N)) = F ((x(N)) ϕ)

and the same holds for ϕn.We need the next lemma.

Lemma 5.6.1. Let L be a Levy process with Levy measure ν and let T > 0. Thenfor each K > 0 there exist constants b and σ such that for each stopping time Rwith |∆LR−| < K we have∥∥∥∥∫ ·

0

H(s−) dLR−(s)∥∥∥∥2

H2[0,T ]

≤ 2

(b2T + 2σ2 + 2

∫(−K,K)

u2 ν(du)

)∫ T

0

EH(t)2 dt,

for every predictable process H with∫ T

0EH(t)2 dt <∞.

Proof. Let K > 0 and let R be a stopping time such that |∆LR−| < K, and letH be a predictable process such that

∫ T0

EH(t)2 dt < ∞. Consider the Levy-Itodecomposition of L (see [9, Theorem 2.4.16]),

L(t) = bt+ σB(t) +∫

(−K,K)

uN(t,du) +∫|u|≥K

uN(t,du)

and set LK(t) = L(t)−∫|u|≥K uN(t,du). Then LR− = LR−K , so we have∥∥∥∥∫ ·

0

H(t) dLR−∥∥∥∥H2[0,T ]

=∥∥∥∥∫ ·

0

H(t) dLR−K

∥∥∥∥H2[0,T ]

≤ 2∥∥∥∥∫ ·

0

H(t−) dLK

∥∥∥∥H2[0,T ]

,

argueing as in [87, Proof of Theorem V.5]. Denote

IH(t) =∫ t

0

∫(−K,K)

uH(t) N(dt,du),

where the integral is as defined in [9, Section 4.2]. Notice that IH(t) equals theusual stochastic integral of H with respect to the Levy process

∫(−K,K)

uN(t, du),as one can see from the construction of both integrals. Since we have that

196

Section 5.6

∫ T0

∫(−K,K)

E(uH(t))2 dtν < ∞ by assumption and Fubini, [9, Theorem 4.2.3]yields that IH is a square integrable martingale with

EIH(t)2 =∫

(−K,K)

u2dν∫ T

0

EH(t)2 dt.

Now ∫ t

0

H(s) dLK(s) = b

∫ t

0

H(s) ds+ σ

∫ t

0

H(s)dB(s) + IH(t),

where the first term is a process of bounded variation and the latter two termsare square integrable martingales. Hence by a well known identity for squareintegrable martingales (see [87, Cor. 3 to Theorem II.27]),∥∥∥∥∫ ·

0

H(s) dL(s)∥∥∥∥2

H2[0,T ]

≤ 2E(TV

(b

∫ ·0

H(s) ds))2

+ 2E

(σ

∫ T

0

H(s) dB(s) + IH(t)

)2

≤ 2b2T∫ T

0

EH(t)2 dt+ 4σ2

∫ T

0

EH(t)2 dt+ 4E(IH(T ))2.

Theorem 5.6.2. Let Assumption 1 and (5.6.1) hold. If Φn → Φ in D[−α, 0]w.r.t. dβ in probability, then XΦn → XΦ uniformly on compact subintervals of[0,∞) in probability.

Proof. Write X = XΦ, Xn = XΦn throughout this proof. Fix T > 0 and ε > 0.Assume first that Φn,Φ is a uniformly bounded family. Hence Theorem 5.5.1

can be applied and (5.5.1) holds, so that there exists N0 such that for N > N0,

supn

P(supt≤T|Xn(t)| > N),P(sup

t≤T|X(t)| > N) < ε.

Define stopping times Tn := inft : |Xn(t)| > N0 and T∞ := inft : |X(t)| >N0. Then P(Tn > T ) > 1− ε and P(T∞ > T ) > 1− ε. Moreover,

(Xn)Tn−(t) = Φn(0) +

∫ t∧Tn−

0

∫[−α,0]

(Xn)Tn−(s+ a)µ(da) ds

+∫ t

0

F ((Xn)Tn−)(s−) dLT

n−(s)

= Φn(0) +∫ t∧Tn−

0

∫[−α,0]

(Xn)Tn−(s+ a)µ(da) ds

+∫ t

0

FN0((Xn)Tn−)(s−) dLT

n−(s).

197


delay equations

So by uniqueness of solutions, (Xn)Tn− = (X(N0)

Φn)T

n−, where X(N0)Φn

denotes thesolution to the equation (5.1.1) with F replaced by FN0 . Likewise, (X)T

∞− =(X(N0)

Φ )T∞−. Since

supt≤T|Xn(t)−X(t)| ≤ sup

t≤T|Xn(t)−X(N0)

Φn(t)|+ sup

t≤T|X(N0)

Φn(t)−X(N0)

Φ (t)|

+ supt≤T|X(N0)

Φ (t)−X(t)|,

we obtain for δ > 0,

P(supt≤T|Xn(t)−X(t)| > δ) ≤ P(sup

t≤T|Xn(t)−X(N0)

Φn(t)| > δ/3)

+ P(supt≤T|X(N0)

Φn(t)−X(N0)

Φ (t)| > δ/3) + P(supt≤T|X(N0)

Φ (t)−X(t)| > δ/3)

≤ P(Tn ≤ T ) + P(supt≤T|X(N0)

Φn(t)−X(N0)

Φ (t)| > δ/3) + P(T∞ ≤ T )

≤ 2ε+ P(supt≤T|X(N0)

Φn(t)−X(N0)

Φ (t)| > δ/3).

Hence in this special case there is no loss of generality by assuming that F isLipschitz in the sense of [92, (2.5)].

Let R be a stopping time such that LR− has bounded jumps, is α-sliceablefor suitably small α, and P(R > T ) > 1 − ε (see [87, Theorem V.5]). Denote byZ and Zn the solutions of equation (5.1.1) with L replaced by LR− and initialcondition Φ and Φn, respectively. By uniqueness of solutions, (Zn)R− = (Xn)R−

and ZR− = XR−. Hence for δ > 0,

P( sup0≤t≤T

|Xn(t)−X(t)| ≥ δ)

≤ P( sup0≤t≤T

|Xn(t)−X(t)| ≥ δ and R ≥ T ) + P(R < T )

≤ P( sup0≤t≤T

|Zn(t)− Z(t)| ≥ δ) + ε/2,

so it suffices to show that Zn → Z uniformly on [0, T ] in probability. To show thiswe introduce some notation:

Y n(t) :=∫ t

0

∫[−α,0]

Z Φ(s+ u)− Z Φn(s+ u)µ(du)ds

+∫ t

0

(F (Z Φ)− F (Z Φn))(s−) dLR−(s),

Gn(U)(t) :=∫

[−α,0]

(Z Φn(t+ u)− (Z − U) Φn(t+ u))µ(du),

Hn(U)(t) := F (Z Φn)(t)− F ((Z − U) Φn)(t),

198

Section 5.6

t ≥ 0. We obtain for Un := Z − Zn the equation

Un(t) = Φ(0)− Φn(0) + Y n(t) +∫ t

0

Gn(Un)(s) ds+∫ t

0

Hn(Un)(s−) dLR−(s).

Lemma V.3.2 of [87] extended to two driving semimartingales yields ‖Un‖S2[0,T ] ≤C‖Φ(0)−Φn(0)+Y n‖S2[0,T ] with a constant C > 0 depending on the process LR−

and the uniform bound for the Lipschitz constants of mappings V 7→∫

[−α,0]V Φ(·+

u)µ(du), V 7→∫

[−α,0]V Φn(·+ u)µ(du), V 7→ F (V Φ), and V 7→ F (V Φn). (Notice

that this boun is finite as F is assumed to be Lipschitz and we assumed thatΦn,Φ is a uniformly bounded family. As Φn → Φ w.r.t. dβ in probability, we haveΦn(0)→ Φ(0) in probability, hence ‖Un‖S2[0,T ] → 0 if ‖Y n‖S2[0,T ] → 0.

Next we show ‖Y n‖S2[0,T ] → 0. Due to the continuous embedding of H2[0, T ]

into S2[0, T ] (see [87, Theorem V.2]) and Lemma 5.6.1 this follows if

E∫ T

0

(∫[−α,0]

(Z Φ(t+ u)− Z Φn(t+ u))µ(du)

)2

dt

+ E∫ T

0

(F (Z Φ))(t)− F (Z Φn)(t))2 dt→ 0

(5.6.2)

as n→∞. Due to the boundedness of F and assumption (5.6.1) the convergenceE∫ T

0(F (Z Φ)(t)− F (Z Φn)(t))2 dt→ 0 as n→∞ holds. Moreover,

∫ T

0

(∫[−α,0]

Z( Φ(t+ u)− Z Φn(t+ u))µ(du)

)2

dt

=∫ T

0

(∫[−α,0]

(Φ(t+ u)− Φn(t+ u))µ(du)

)(∫

[−α,0]

(Φ(t+ v)− Φ(t+ v))µ(dv)

)dt

=∫

[−α,0]

∫[−α,0]

∫ T

0

(Φ(t+ u)− Φn(t+ u))(Φ(t+ v)− Φn(t+ v))1[−α,−t)(u)

· 1[−α,−t)(v) dtµ(du)µ(dv).

This expression converges almost surely to zero and is bounded in n and ω, asconvergence in dβ implies almost everywhere convergence on [−α, 0] and the familyΦn,Φ is uniformly bounded. Hence (5.6.2) holds indeed, and we proved the specialcase of uniformly bouded initial conditions.

For the general case, notice that since Φn → Φ w.r.t. dβ in probability, the lawsof Φn converge weakly to the law of Φ, and since (D[−α, 0], dβ) is Polish we have bythe Prohorov theorem that the family of laws of Φn,Φ is tight. Hence for a ε > 0

199


delay equations

there is a set K ⊂ D[−α, 0] compact w.r.t. dβ such that P(Φn ∈ K, Φ ∈ K) > 1−εfor all n. As convergence w.r.t. dβ is implied by Skorokhod convergence, K is alsocompact w.r.t. dS . Hence all the functions in K are bounded by some finiteconstant C. Consider the truncated initial conditions ΦCn and ΦC and let X

nand

X be the solutions of equation (5.1.1) with these initial conditions. We have thatP(Φn = ΦCn , Φ = ΦC) > 1 − ε, and concentrating P on the sets Φn = ΦCn andΦ = ΦC, with the aid of [87, Theorem IV.23] and the uniqueness of solutionswe conclude that

Xn

= Xn a.s. on Φn = ΦCn X = X a.s. on Φ = ΦC

Moreover, it is easy to check that ΦCn → ΦC w.r.t. dβ in probability, so that bethe special case above, sup0≤t≤T |X

n(t) − X(t)| → 0 in probability. Finally, for

δ > 0,

P( sup0≤t≤T

|X(t)−Xn(t)| > δ) ≤ P( sup0≤t≤T

|X(t)−X(t)| > δ/3)

+ P( sup0≤t≤T

|X(t)−Xn(t)| > δ/3) + P( sup

0≤t≤T|Xn

(t)−Xn(t)| > δ/3)

≤ 2ε+ P( sup0≤t≤T

|Xn(t)−Xn(t)| > δ/3)

and the theorem has been proved.

Remark 5.6.3. 1. By stopping X = XΦ appropriately, we can use similartechniques as before and prove Theorem 5.6.2 even if F is not bounded, butmerely having linear growth.

2. Each of the conditions ‘Φn → Φ w.r.t. dS in probability’, ‘Φn → Φ w.r.t. dβa.s.’, and ‘Φn → Φ w.r.t. dS a.s.’ is stronger than the condition of Theorem5.6.2.

In the next Corollary the use of dβ instead of dS is essential; see [92, Section3.3] for a counterexample with dS .

Corollary 5.6.4. Let Assumption 1 and (5.6.1) hold. If Φn → Φ with respect todβ in probability, then XΦn,t → XΦ,t with respect to dβ in probability for everyt ≥ 0.

Proof. If t ≥ α, the assertion readily follows from Theorem 5.6.2. Consider 0 <t < α. Let λ be an increasing homeomorphism from [−β, 0] onto itself. Defineρ : [−β, 0]→ [−β, 0] by ρ(s) = s for s ∈ [−t, 0], ρ(s) = λ(t+s)− t for s ∈ [−α,−t),and affine on [−β,−α] with ρ(−β) = −β. Then ρ is an increasing homeomorphism,sups∈[−β,0] |ρ(s)− s| ≤ sups∈[−β,0] |λ(s)− s|, and

sups∈[−α,0]

|XΦn(t+ s)−XΦ(t+ ρ(s))| ≤ sups∈[−α,−t]

|Φn(t+ s)− Φ(λ(t+ s))|

∨ sups∈[−t,0]

|XΦn(t+ s)−XΦ(t+ s)|.

200

Section 5.7

Hence dβ(XΦn,t, XΦ,t) ≤ dβ(Φn,Φ) + sups∈[0,α] |XΦn(s) − XΦ(s)|, and the prooffollows.

5.7 Markov and eventual Feller property and ex-istence of invariant measure

Theorem 5.7.1. Let Assumption 1 and (5.6.1) hold. Then the segment process(XΦ,t)t≥0 is a Markov process, the transition operators Ps,t defined by (5.4.3) mapBd(D[−α, 0]) into Bd(D[−α, 0]) and satisfy

Pu,t = Pu,sPs,t and Ps,t = P0,t−s

for all 0 ≤ u ≤ s ≤ t. Moreover, the Markov semigroup (Pt)t≥0 defined by (5.4.4)is eventually Feller.

Proof. In this proof we endow D[−α, 0] with the metric dβ . Recall that by Lemma5.2.1 dβ and dS generate the same Borel σ-algebra B(D[−α, 0]).

We begin by showing (5.4.2). Let 0 ≤ u ≤ t and B ∈ B(D[−α, 0]). Observethat 1B(Xu

XΦ,t−u) is measurable, as XuXΦ,t−u = XΦ(t), by (5.4.1). Let Cb denote

the space of bounded functions f : D[−α, 0]→ R that are continuous with respectto dβ . Let f ∈ Cb and let ξ be an Fu-measurable random variable with valuesin D[−α, 0]. For a Fu-measurable random variable ξ with values in D[−α, 0], letA(ξ, ω) := f(Xu

ξ,t(ω)). Let further A(ϕ) := EA(ϕ, ·) for ϕ ∈ D[−α, 0]. Assumefirst that

ξ =n∑i=1

ai1Ci (5.7.1)

with ai ∈ D[−α, 0] and Ci ∈ Fu, and Ci mutually disjoint,⋃i Ci = Ω, and

P(Ci) > 0 for all i. Then A(ξ, ω) =∑iA(ai, ω)1Ci(ω) (as before we rescale P to

Ci and use [87, Theorem IV.23] and uniqueness of solutions), so

E[A(ξ(·), ·)|Fu] =∑i

1CiEA(ai, ·) =∑i

A(ai)1Ci = A(ξ(·)).

If ξ is an arbitrary Fu-measurable random variable with values in D[−α, 0], thenthere are ξm of the form (5.7.1) such that dβ(ξm(ω), ξ(ω))→ 0 as m→∞ for a.e.ω (see [105, Proposition I.1.9]). Due to the continuity of f and Corollary 5.6.4 wehave A(ξm, ·)→ A(ξ, ·) in probability, so that

A(ξm) = E[A(ξm, ·)|Fu]→ E[A(ξ, ·)|Fu],

as f is bounded. Again by Corollary 5.6.4 we have A(ϕn) → A(ϕ) wheneverdβ(ϕn, ϕ)→ 0, so that A(ξm)→ A(ξ) a.s. Hence E[A(ξ, ·)|Fu] = A(ξ) a.s.

Next let C be a closed subset of D[−α, 0] and choose fn ∈ Cb such thatfn ↓ 1C pointwise. Let An(ω) := fn(Xu

ϕ,t(ω)) and A(ϕ, ω) = 1C(Xuϕ,t(ω)), ω ∈ Ω,

201


delay equations

and An(ϕ) = EAn(ϕ, ·) and A(ϕ) = EA(ϕ, ·), ϕ ∈ D[−α, 0]. Then An ↓ A andAn ↓ A pointwise, so

E[A(ξ, ·)|Fu] = limn→∞

E[An(ξ, ·)|Fu] = limn→∞

An(ξ) = A(ξ) a.s.

By a monotone class argument we can extend the above identity to any C ∈B(D[−α, 0]), that is, we have proved (5.4.2).

We show that Ps,t maps Bb(D[−α, 0]) into Bb(D[−α, 0]). Indeed, if f ∈ Cb,then Corollary 5.6.4 yields that Ps,tf ∈ Cb. If C is a closed subset of D[−α, 0],then there are fn ∈ Cb such that fn ↓ 1C pointwise and then Ps,tfn ↓ Ps,t1Cpointwise, so Ps,t1C ∈ Bb(D[−α, 0]). By a monotone class argument we obtainPs,t1C ∈ Bb(D[−α, 0]) for any F ∈ B(D[−α, 0]) and then it follows that Ps,tf ∈Bb(D[−α, 0]) for any f ∈ Bb(D[−α, 0]).

The Markov property (5.4.2) yields for 0 ≤ u ≤ s ≤ t that

Pu,tf(ϕ) = Ef(Xuϕ,t−u) = E(E[f(Xu

ϕ,t−u)|Fs])= E(E[f(Xs

Xuϕ,s−u,t−s)|Fs]) = E(Ef(Xsψ,t−s)|Xuϕ,s−u=ψ)

= Pu,sPs,tf(ϕ).

By uniqueness in law [53, Subsection IX.6c] we have that (Xuϕ,t)t≥0 has the same

law as (Xϕ,t)t≥0 for each u ≥ 0, since Lu and L have the same law. HencePs,t = P0,t−s.

Finally, we establish that (Pt)t is eventually Feller. By Proposition 5.6.2 wehave that for each t ≥ α, ϕn → ϕ in D[−α, 0] implies Xϕn,t → Xϕ,t in D[−α, 0]in probability, so Ptf(ϕn) → Ptf(ϕ), hence (5.4.5) holds. Property (5.4.6) hasalready been shown.

Remark 5.7.2. The condition (5.6.1) is rather mild as the following examplesshow.

1. Let ρ be a finite signed Borel measure on [−α, 0] and let f be a locallyLipschitz function on R with linear growth. Define

F (x)(t) := f

(∫[−α,0]

x(t+ v)ρ(dv)

), t ≥ 0,

F (x)(t) := 0, −α ≤ t ≤ 0,

202

Section 5.7

x ∈ D[−α,∞). Then if ϕn → ϕ in D[−α, 0] and x ∈ D[0,∞),∫ α

0

(F (x ϕ(t))− F (x ϕn(t)))2 dt

=∫ α

0

(f

(∫[−α,0]

x ϕn(t+ v)ρ(dv)

)− f

(∫[−α,0]

x ϕ(t+ v)ρ(dv)

))2

dt

≤ C∫ α

0

(∫[−α,0]

(x ϕn(t+ v)− x ϕ(t+ v))ρ(dv)

)2

dt

= C

∫ α

0

(∫[−α,0]

(x ϕn(t+ v)− x ϕ(t+ v))ρ(dv)

)

·

(∫[−α,0]

(x ϕn(t+ u)− x ϕ(t+ u))ρ(du)

)dt

= C

∫ α

0

∫[−α,0]

∫[−α,0]

(ϕn(t+ v)− ϕ(t+ v)) (ϕn(t+ u)− ϕ(t+ u))

· 1u<−t1v<−tρ(du)ρ(dv) dt,

for some C depending only on f , x, ρ, and (ϕn), where the latter equalityfollows by Fubini theorem and the fact that x ϕn(t+w) = x ϕ(t+w) whenevert+w ≥ 0. Now by the Fubini theorem and dominated convergence F satisfies(5.6.1) for each x ∈ D[0,∞).

Moreover, F is lolidet which can be seen as follows: for t ≥ 0 and x, y ∈D[−α,∞) such that sups∈[1−α,t] |x(s)| ∨ |y(s)| ≤ n we have∣∣∣∣∣

∫[−α,0]

x(t+ v) ρ(dv)

∣∣∣∣∣ ,∣∣∣∣∣∫

[−α,0]

y(t+ v) ρ(dv)

∣∣∣∣∣ ≤ n|ρ|.Hence as f is locally Lipschitz there is a Cn > 0 such that

|F (x)(t)− F (y)(t)| ≤ Cn

∣∣∣∣∣∫

[−α,0]

(x(t+ v)− y(t+ v)) ρ(dv)

∣∣∣∣∣≤ Cn|ρ| sup

v∈[−α,0]

|x(t+ v)− y(t+ v)|.

Since f has linear growth, it follows that F is lolidet.

However, F need not be Lipschitz in the sense of [92, (2.5)] if f is notLipschitz. To see this, take f(t) = sin(t2), ρ the Lebesgue measure on[−α, 0], and evaluate F (xn)(t) = sin(α2n2), where xn ≡ n.

2. Likewise we can take ρ1, . . . , ρd signed Borel measures on [−α, 0] and f alocally Lipschitz on Rd. In particular, we may take for F combinations offinitely many point evaluations.

As above, F is lolidet but need not be Lipschitz in the sense of [92, (2.5)].

203

Section 5.7

a Cn > 0 such that

|F (X)(t)− F (y)(t)| ≤ Cn

∣∣∣∣∣ sup[t−α,t]

|x(s)| − sup[t−α,t]

|y(s)|

∣∣∣∣∣ .As sups∈[t−α,t] |x(s)| ≤ sups∈[t−α,t] |x(s)−y(s)|+sups∈[t−α,t] |y(s)|, we obtain

by symmetry∣∣∣sups∈[t−α,t] |x(s)| − sups∈[t−α,t] |y(s)|

∣∣∣ ≤ sups∈[t−α,t] |x(s) −y(s)|. Since f has linear growth, it follows that F is lolidet. Again, F neednot be Lipschitz in the sense of [92, (2.5)] if f is not Lipschitz, as we see bytaking f(t) = sin(t2) and evaluating F on the sequence xn ≡ n.

4. Similar arguments as in 3. apply to functionals like f(supt−α≤s≤t x(s)),f(inft−α≤s≤t x(s)) and f(inft−α≤s≤t |x(s)|).

Notice that all functionals F in the previous remark are autonomous in thesense of Definition 5.4.4. If f is bounded, then F satisfies all conditions of As-sumption 1 and (5.6.1).

Finally we consider existence of an invariant measure. Denote by P the setof Borel probability measures on D[−α, 0] endowed with the topology of weakconvergence of measures. Let Bb denote the space of all real valued boundedBorel functions on D[−α, 0] and denote 〈ζ, f〉 =

∫f dζ, f ∈ Bb, ζ ∈ P. The

adjoint of the Markov semigroup defined in (5.4.4) is given by

〈P ∗t ζ, f〉 = 〈ζ, Ptf〉, f ∈ Bb, ζ ∈ P.

A measure η ∈ P is called an invariant measure for (5.1.1) if

P ∗t η = η for all t ≥ 0.

If η is the distribution of an initial segment Φ, then P ∗t η is the distribution of thesegment Xt,Φ. Therefore if Φ is an F0-measurable random variable with values inD[−α, 0] whose law is an invariant measure, the segment process corresponding tothe solution X of (5.1.1) with initial condition Φ is constant in law. In this casethe solution X itself is also constant in law.

Theorem 5.7.3. Grant Assumption 1 and assume that (5.6.1) holds for everyx ∈ D[0,∞). Then equation (5.1.1) has an invariant measure.

Proof. It follows from Theorem 5.7.1 that Pt maps Cb = Cb(D[−α, 0]) into Cbfor t ≥ α and that t 7→ P ∗t ζ is a continuous map from [α,∞) to P. Moreover,P ∗s+t = P ∗s P

∗t for all s, t ≥ 0. Theorem 5.5.1 yields that the set P ∗t ζ : t ≥ α is

tight, where, for instance, ζ is the distribution of the initial condition ϕ ≡ 0.Next, proceeding as in [92, Section 4.2], the invariant measure η is obtained by

means of the Krylov-Bogoliubov method.

Example 5.7.4. Let us illustrate by means of an example how transformation ofan equation with unbounded globally Lipschitz coefficients may lead to an equation

205


delay equations

with bounded coefficients that are locally but not globally Lipschitz. Consider theequation

dX(t) = −ax(t)dt+ f(Xt)dt+ g(Xt)dW (t), t ≥ 0,where a > 0, α > 0, and f, g : D[−α, 0] → R are Lipschitz with respect to ‖·‖∞and continuous with respect to the Skorohod metric dβ and such that

|f(y)| ∨ |g(y)| ≤ C(1 + |y(0)|r) for all y ∈ D[−α, 0]

for some C > 0 and 0 < r < 1. The process W is a real valued Wiener process.With a sufficiently smooth strictly increasing φ : R→ R such that

φ(x) = sgn(x)|x|s for large |x|

for some 0 < s < 1− r, the process

Y (t) = φ(X(t))

satisfiesdY (t) = −asY (t)dt+ f(Yt)dt+ g(Yt)dW (t),

due to Ito’s formula. The coefficients f and g are given by

f(y) = asy(0)− asφ′(φ−1(y(0)))φ−1(y(0)) + φ′(φ−1(y(0)))f(t 7→ φ−1(y(t)))

+12φ′′(φ−1(y(0)))g(t 7→ φ−1(y(t)))2,

g(y) = φ′(φ−1(y(0)))g(t 7→ φ−1(y(t))),

for y ∈ D[−α, 0]. A computation reveals that f and g are bounded. Indeed, chooseR ≥ 1 such that φ(x) = sgn(x)|x|s for |x| ≥ R1/s. Then φ′(φ−1(x))φ−1(x) = sx,|φ′(φ−1(x))| = s|x|1−1/s, and |φ′′(φ−1(x))| = s(1− s)|x|1−2/s for |x| ≥ R, so that

|φ′(φ−1(x))|(1 + |x|r/s) ≤ 2s and |φ′′(φ−1(x))|(1 + |x|r/s)2 ≤ 4s(1− s)

for |x| ≥ R and therefore

|f(y)| ≤ |asy(0)− aφ′(φ−1(y(0)))φ−1(y(0))|+ C|φ′(φ−1(y(0)))|(1 + |φ−1(y(0))|r)

+12C|φ′′(φ−1(y(0)))|(1 + |φ−1(y(0))|r)2

≤M ∨ 2C(s+ s(1− s))

for all y ∈ D[−α, 0], where

M = sup|x|≤R

|asx− aφ′(φ−1(x))φ−1(x)|+ C|φ′(φ−1(x))|(1 + |φ−1(x)|r)

+12C|φ′′(φ−1(x))|(1 + |φ−1(x)|r)2,

and, similarly,|g(y)| ≤M ∨ 2s for all y ∈ D[−α, 0].

It easily follows from the local Lipschitz continuity of f , g, φ′, φ′′ and φ−1 that fand g are locally Lipschitz continuous. Since φ−1 is not Lipschitz, f and g neednot be globally Lipschitz, even if f and g are globally Lipschitz.

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Curriculum vitae

Igor Stojkovic was born in August of 1972, in Belgrade. In 1991 he completed hishigh school education at the Matematicka Gimnazija in Belgrade. This extraor-dinary institution is well known far beyond the borders of the Republic of Serbia,due to its special education programs and the frequent high achievements of itsstudents at the International Olympiad competitions in Mathematics and Physics.A remarkably high proportion of its former students were successful in academiaand quite a number of them became professors at well established universitiesworld wide.

In 1991 he started his study of mathematics at the Belgrade University, butafter having completed only one semester, due to the outbreak of the civil war informer Yugoslavia, he fled the country and arrived in The Netherlands in 1992.After eight long years of uncertainty, he obtained his Dutch residence permit onhumanitarian grounds in 2000. After passing his state exam of proficiency in Dutchlanguage, he started computer science studies at the University of Amsterdam(UvA) in the Fall of 2000. Between September 2001 and February 2007 he studiedmathematics at the UvA.

He graduated summa cum laude in Mathematics in February 2007 at the Uni-versity of Amsterdam. His master thesis advisor was Dr. Peter Spreij, and thegoal of his master thesis was to investigate which conditions guarantee that thestochastic convolutions with a deterministic integrand possess the semimartingaleproperty, a problem proposed by Dr.ir. Onno van Gaans, who also acted as acoadvisor.

Igor was appointed as a PhD student at the Mathematical Institute of LeidenUniversity in February 2007, on the VIDI grant “Stationary dynamics in infinitedimensions” of Onno van Gaans funded by the Netherlands Organisation for Sci-entific Research (NWO). Professor Dr. Sjoerd Verduyn Lunel acted as advisor andOnno van Gaans as coadvisor. The first research that Igor Stojkovic did as agraduate student focused on proving the existence of an invariant measure for aclass of stochastic delay differential equations under a locally Lipschitz condition.During the same initial period of his appointment at Leiden University, he hasgained interest in the new developing field of gradient flows in metric spaces andthe optimal transportation theory. Upon a research proposal by Professor Dr.Philippe Clement, he shifted the epicenter of his research to developing severalfundamental extensions of the theory of gradient flows in the context of CAT(0)

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and Wasserstein spaces.He has also contributed to the education activities of the Mathematical Insti-

tute of Leiden University, in particular, by assisting Dr. Flora Spieksma in teach-ing the Dutch national master course “Stochastic processes” for three consecutiveyears.

The author of the thesis has visited a number of conferences and workshopson stochastic analysis and optimal transportation theory. He paid a research visitto Professor Dr. Giuseppe Savare at Pavia University in 2010. Also, upon aninvitation by Professor Luigi Ambrosio of Scuola Normale Superiore in Pisa, hewas a research visitor of the Ennio De Giorgi Insitute in 2010. Furthermore, hevisited Professor Dr. Genaro Lopez at Sevilla University , and also Professor Dr.Anton Petrunin at the University of Munster. Currently, Igor Stojkovic holds apostdoc position at Delft University of Technology under supervision of ProfessorDr. Jan van Neerven.

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