Geometric approach to evolution problems in metric spaces

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Geometric approach to evolution problems in metric spaces

Proefschrift

ter verkrijging van

de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus

prof. mr. P.F. van der Heijden,

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

klokke 15:00 uur

door Igor Stojkovi´ c geboren te Belgrado

in 1972

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promotor: prof.dr. S.M. Verduyn Lunel copromotor: dr.ir. O.W. van Gaans

overige leden: prof.dr. L. Ambrosio (Scuola Normale Superiore, Pisa, Itali¨ e)

prof.dr. J.M.A.M. van Neerven (Technische Universiteit Delft)

prof.dr. M.A. Peletier (Technische Universiteit Eindhoven)

prof.dr. P. Stevenhagen

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Geometric approach to evolution

problems in metric spaces

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Artwork on the cover designed by Brett Daniel, www.BrettDaniel.com Igor Stojkovi´c, Leiden, 2011c

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Acknowledgment

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

I thank my advisors Sjoerd Verduyn Lunel and Onno van Gaans for granting me freedom in choosing my own research topics. I also thank them for having confidence in me and for supporting me morally regarding my decision to step into the new field of gradient flows in metric spaces. I specially thank Onno van Gaans for his careful checking of my manuscript and many suggestions regarding improvement of many of my introductory texts. Onno hes helped me a lot to add a nice touch to my manuscript, which has really made it a better thesis. Contrary to his conviction that he only did job in this regard, I think that some of his input has 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 work needed to be done. At times we worked together on polishing it far beyond the office hours. I find it extraordinary kind of him to help me in this way, and I can not thank him enough for it.

I thank Professer Philippe Cl´ement for his seminars on gradient flows in metric spaces, which have introduced me into this beautiful field. I appreciate that he proposed me a problem which invoked my engagement in this field.

I thank Professor Giuseppe Savar´e and Professor Luigi Ambrosio for inviting me to their institutes and for interesting discussions from which I learned new things. I moreover thank Professor Ambrosio for his kind accepting of the task of being a member of the commission which has approved this thesis, and for his advice regarding my publications.

I would like to express my gratitude to Anton Petrunin for his hospitality during my visit at University of M¨unster, and for his generous sharing of his mathematical insights which helped me to improve my results of Chapter 2.

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

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

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

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difficult times.

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Samenvatting

In dit proefschrift presenteert de auteur de resultaten van het onderzoek uitgevoerd tijdens zijn aanstelling als promovendus aan de Universiteit Leiden. Vier essentieel verschillende onderwerpen zijn bestudeerd en de verkregen resultaten kunnen ge- kenmerkt worden als uitbreiding van de bestaande theorie voor gradi¨entstromingen op metrische ruimten. Een belangrijk uitgangspunt zijn gradi¨entstromingen op Wasserstein-2 ruimten van kansmaten op Euclidische ruimte R^d.

De theorie van gradi¨entstromingen kan gezien worden als een onderdeel van de theorie van optimaal transport. Deze theorie staat zeer in de belangstelling en in het afgelopen decennium zijn er een aantal belangrijke doorbraken gerealiseerd.

Een belangrijk streven in de studie van de gradiëntstromingen op verschillende metrische ruimten, is het vinden van tegenhangers van de elegante theorie van de variationele analyse op lineare ruimten. Een wezenlijk deel van deze tegenhangers is geconstrueerd door Ambrosio-Gigli-Savaré in de monografie [5]. In dit werk maken de auteurs op een essentiële manier gebruik van specifieke meetkundige eigenschappen van de Wasserstein-2 ruimten over Hilbertruimten, meer specifiek van de gegeneraliseerde convexiteit van de functie W₂². Een ander belangrijk streven in de theorie van gradiëntstromingen op metrische ruimten is het vinden van nieuwe toepassingen van de abstracte theorie en, in het bijzonder, het vinden van partiële differentiaalvergelijkingen die in zekere zin als dergelijke gradiëntstroming ge¨ınterpreteerd kunnen worden.

In zijn onderzoek heeft de auteur zijn aandacht gericht op zowel de uitbreiding van de bestaande abstracte theorie van de gradiëntstromingen op metrische ruimten (in twee verschillende richtingen), alsmede op een behandeling van een klasse van partiële differentiaalvergelijkingen die niet kunnen worden ge¨ınterpreteerd als een gradiëntstroming op een Wasserstein-2 ruimte, maar wel in een verwante context met de Wasserstein-2 ruimte als toestandsruimte.

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

n→∞lim(e⁻ⁿ^t^Ae⁻ⁿ^t^B)ⁿ= e^−(A+B), ∀t ∈ R, (0.0.1) voor alle matrices A, B ∈ R^d×d. Het verband tussen formule (0.0.1) en de productformule op Hilbertruimten, en algemener op de CAT(0)-ruimten, is dat voor

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elke matrix A ∈ R^d×d en voor elke x ∈ R^d de kromme 0 6 t 7→ e^−tAx de unieke oplossing is van de differentiaalvergelijking

d

dtx(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 de oplossingen van differentiaalvergelijkingen met respectievelijk de matrices A en B als generatoren. Zo’n uitspraak is, zodra twee gradi¨entstromingshalfgroepen gegeven zijn, in een algemenere puur metrische context gemakkelijk te formuleren.

Het is echter geen eenvoudige zaak of de gegeneraliseerde Trotter-Kato formule in zo’n algemene context nog geldt.

De primaire doelstelling van het onderzoek van de auteur gepresenteerd in Hoofdstuk II is om de natuurlijke analogie van de productformules in het kader van gradi¨entstromen op CAT (0)-ruimten te bewijzen. CAT(0)-ruimten zijn metrische ruimten met niet-positieve kromming in de zin van Alexandrov. Vanuit nu- merieke approximatie gezien zijn de versies van productformules waar resolventen worden gebruik meer interessant dan die geformuleerd in termen van halfgroepen.

Het bewijs van de productformules in Hilbertruimten vereist een aantal andere approximatiestellingen en specifiek wiskundig gereedschap met betrekking tot in- tegratie van krommen en een ongelijkheid zoals die van Gronwall. Tegenhangers van 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 dat het bewijs meer direct wordt. Technieken die in Hoofdstuk II worden gebruikt komen uit de theorie van metrische meetkunde, gradi¨entstromingen op metrische ruimten, en evolutie vergelijkingen op Banach ruimten. Gebrek an een bevredigende analogie van het concept van de zwake convergentie op CAT(0 ruimten heeft een forse belemering gegeven. Waar in de bewijzen van de productformules op Hilbertruimten het concept van zwakke convergentie gebruikt wordt, worden in het algemenere CAT(0)-geval ultra-limieten, ultra-producten en ultra-extensie technieken successvol gebruikt.

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

∂tρt= ∆xρt+ ∇ · (bρt), in D⁰((0, +∞) × R^d) (0.0.3) (waar b : R^d → R^d een monotone afbeelding is, maar niet noodzakelijk een gradiënt van een convexe functie) een contractieve halfgroep op de Wasserstein-2 ruimte (P2(R^d), W2) induceert, die bovendien dezelfde padregulariteitseigenschap- pen heeft als de halfgroepen van gradiëntstromingen (bijvoorbeeld lokaal Lip- schitz in de tijdvariabele). Terwijl de halfgroepen ge¨ınduceerd door oplossingen van de symmetrische versies van de Fokker-Planck vergelijkingen wel gradiënt- stromingshalfgroepen zijn, kunnen de oplossingen van de niet-symmetrische Fok- ker-Planck vergelijkingen niet ge¨ınduceerd worden door een convexe functionaal op (P₂(R^d), W₂). Deze constatering impliceert dat de bestaande theorie niet al- gemeen genoeg is om de vergelijkingen (0.0.3) te analyseren, althans niet door

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zoek naar de niet-symmetrische Fokker-Planck vergelijkingen kwam op gang nadat de auteur had bewezen dat de Trotter-Kato productformule voor de symmetrische versies van de Fokker-Planck vergelijkingen geldt (dat wil zeggen wanneer b wel een gradi¨ent is), ten opzichte van de Wasserstein-2 metriek. Dezelfde methode kan gebruikt worden om de stroming van de oplossingen van de niet-symmetrische Fokker-Planck vergelijkingen ook als een halfgroep op (P2(R^d), W2) te beschou- wen en te bewijzen dat de halfgroep in deze zin contractief is. Verder wordt in Hoofdstuk III bewezen dat de paden van deze halfgroep lokaal absoluut continu zijn en zelfs dat het zogenaamde regulariserende effect optreedt. Het geheel van de verkregen resultaten bevestigt dat de meetkundige structuur van de Wasserstein-2 ruimten inderdaad goed verenigbaar is met de structuur van de niet-symmetrische Fokker-Planck vergelijking. Anders gezegd, de Waserstein-2 ruimte is inderdaad een natuurlijke omgeving voor de niet-symmetrische Fokker-Planck vergelijking.

Bovendien, als men de hoodfstelling van Hoofdstuk III met de hoodfstelling over de gradi¨entstromingen (toegepast op de symmetrische Fokker-Planck vergelijking) vergelijkt, kan men een analogie vaststellen met de verhouding in Hilbertruim- ten van de gradi¨entstromingen tot de stromingen die ge¨ınduceerd worden door maximale monotone operatoren.

Hoofdstuk IV besteedt weer aandacht aan het uitbreiden van de abstracte theorie (althans in eerste instantie), en het doel van de auteur is om een bevredigende theorie van maximale monotone operatoren in een passende gegeneraliseerde zin op te bouwen. Een essentieel uitgangspunt is dat de nieuwe theorie een uitbreiding dient te zijn van de theorie van de gradi¨entstromingen op Wasserstein-2 ruimten.

Een stroming wordt een gradi¨entstroming genoemd als deze wordt ge¨ınduceerd door functionalen die onderhalfcontinu zijn en convex langs gegeneraliseerde geodeten. Een basisuitgangspunt van dit onderzoek is dat men gebruik kan maken van de gegeneraliseerde convexiteit van de functie W₂², dezelfde meetkundige ei- genschap van (P₂(R^d), W₂) die door de auteurs van [5] op essenti¨ele wijze gebruikt is in de opbouw van hun theorie.

Hoofdstuk IV is als volgt opgebouwd. Eerst wordt in gegeneraliseerde zin het begrip maximale monotone operatoren op Wasserstein-2 ruimten ge¨ıntroduceerd.

Vervolgens wordt aangetoond dat de Fréchet subdifferentiaal, zoals gedefinieerd door Ambrosio-Gigli-Savaré voor functionalen die convex zijn langs de gegeneraliseerde geodeten, maximale monotone operatoren volgens deze definitie zijn. Het blijkt nu dat dergelijke operatoren en de bijbehorende resolventen een aantal eigenschappen hebben die geheel analoog zijn aan het Hilbertruimte geval. Deze resultaten gelden ook voor gradiëntstromingen, maar waren daarvoor niet eerder in de literatuur bewezen noch geformuleerd. Het onderzoek gepresenteerd in Hoofdstuk IV wordt afgerond met het geven van een bewijs van de hoofdstel- ling over de existentie van oplossingen van het bijbehorende Cauchy-probleem en de fundamentele eigenschappen van de ge¨ınduceerde halfgroep. Het kan worden opgemerkt dat er een conceptueel verschil is tussen ons bewijs en de reeds gepu- bliceerde bewijzen van de stellingen over existentie van oplossingen in het kader van gradiëntstromingen. Het hoofdstuk wordt afgesloten met een discussie over

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

In het laatste Hoofdstuk V wordt onderzoek gepresenteerd waarmee existentie van een invariante maat voor een stochastische differentiaalvergelijking met een tijdvertraging in zowel de drift als in de diffusieterm kan worden bewezen. De drift van de stochastische vergelijking is verondersteld exponentieel stabiel te zijn, en het diffusieproces is een Lévyproces 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 diffusiecoëfficiënt versoepeld is tot een lokale Lipschitzvoorwaarde. De zogenaamde variatie-van- constanten formule is een belangrijk en onmisbaar gereedschap in dit onderzoek en deze formule wordt dan ook bewezen in Hoofdstuk V. Verder wordt een stelling over stabiliteit van de oplossing ten opzichte van de beginvoorwaarde bewezen die ook onafhankelijk van de andere resultaten van belang is. Dit onderzoek is uitgevoerd tijdens de eerste periode van het promotieonderzoek van de auteur, en daarna gebruikt ter voorbereiding van een publicatie in samenwerking met de begeleider Dr. Onno van Gaans. De auteur heeft een bewuste keuze gemaakt om deze resultaaten in het laaste hoofdstuk te presenteren, aangezien de behandelde onderwerp tot de theorie van de stochastische analyse behoort, en niet tot de theorie van optimale transport en gradiëntstromen, het centrale onderwerp van dit proefschrift.

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Chapter 1

Introduction

1.1 Gradient flows, maximal monotone operators and product formulas in Hilbert spaces

1.1.1 Gradient flows

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

d

dtx(t) = −∇ϕ(x(t)), t > 0, x(0) = x0∈ R^d,

(1.1.1)

where x : [0, +∞) → R^d is the unknown function to be solved from the equation.

Its initial value at time t = 0 is an arbitrary but fixed point x₀∈ R^d. Due to our assumption that ϕ be continuously differentiable, one easily argues by means of Picard iteration that (1.1.1) has a unique solution for each x₀∈ R^d. The function ϕ is often called the potential of the gradient flow equation (1.1.1), solutions are usually called gradient flow curves, and the set of all solutions defines the gradient flow associated to the potential ϕ.

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Approximation scheme

The most suitable method for computing the approximate values of the solutions turns out to be the Euler method, i.e. given the initial value x(0) = x0∈ R^d, and a time step size τ > 0, one makes the time scale discrete by restricting the set of observations to times 0, τ, 2τ, 3τ, ... and ‘computes’ the approximate valueex(τ ) by

x(τ ) − x(0)e

τ ≈ −∇ϕ(x(τ ))e (1.1.2)

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

x(τ ) = (I + τ ∇ϕ)e ⁻¹x(0), (1.1.3) where I : R^d → R^d is the identity map. Naturally, the operator (I + τ ∇ϕ)⁻¹ : R^d→ R^din (1.1.3) must be well defined for each τ > 0, and it turns out that the necessary and sufficient condition for this to hold is that ϕ : R^d→ R^d is a convex function, i.e.

ϕ((1 − t)x + ty) 6 (1 − t)ϕ(x) + tϕ(y), ∀ x, y ∈ R^d, ∀ t ∈ [0, 1]. (1.1.4) Since convexity of ϕ implies monotonicity of its gradient ∇ϕ, that is

h∇ϕ(x) − ∇ϕ(y), x − yi > 0, ∀x, y ∈ R^d, (1.1.5) the so called resolvents associated to ∇ϕ, i.e.

J_τ:= (I + τ ∇ϕ)⁻¹ : R^d→ R^d, ∀τ > 0, (1.1.6) are well defined. This claim can be seen to hold as follows. A differentiable function assumes a local extremal value at a point x ∈ R^d, if and only if its gradient is 0 at x. Therefore, for each x ∈ R^d and τ > 0, Jτx must be the unique minimizer of the function

R^d3 y 7→ Φτ(x, y) := 1

2τ|x − y|²+ ϕ(y), (1.1.7) since ∇_yΦ_τ(x, y) = _τ¹(y −x)+∇ϕ(y). Conversely, any minimizer x_τof the function Φτ(x, y) (over y ∈ R^d) solves the implicit equation posed in (1.1.2). With the aid of the convexity assumption on ϕ, it can be easily proven that this minimization problem has a unique solution, hence the operators Jτ in (1.1.6) are well defined for each τ > 0. The approximate valuex(2τ ) of the solutions of (1.1.1) is defined bye ex(2τ ) := (I + τ ∇ϕ)⁻¹x(τ ) = Je τx(τ ) = Je _τ²x(0), and the values of the approximate solution at the subsequent times 3τ, 4τ, ... are given by

ex(kτ ) := (I + τ ∇ϕ)⁻¹ex((k − 1)τ ) = J_τx((k − 1)τ ) = Je _τ^kx(0), k ∈ N. (1.1.8) In order to stress the dependence of the approximate solution on the time step size τ in the approximation, we writexeτ(kτ ), k ∈ N.

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At this point in our analysis several considerations should be given.

The first consideration regards the accuracy of the above discussed approximation procedure. More precisely, if we let τ converge to 0, do our approximate discrete time solutions converge in some way to the unique solution of the abstract Cauchy problem (1.1.1), and if they do what is the order of this convergence? The answer to this question is indeed affirmative, i.e. for each t > 0 and for each x0∈ R^d, we have that

Jt

n

x0−→ 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 convergence is _n¹.

The second consideration regards relaxing our assumption that ϕ : R^d → R is continuously differentiable, to a weaker asumption. In this case we also need to reformulate the Cauchy problem (1.1.1) since the gradient of ϕ appears in it explicitly.

Subdifferential of a convex function

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

ϕ(x) 6 lim inf_n→∞ ϕ(xn) whenever xn→ x, (1.1.10) holds. The gradient ∇ϕ of a C¹ function ϕ then needs to be replaced by the so called subdifferential ∂ϕ of ϕ, which due to the assumed convexity can be defined by

R^d× R^d⊃ ∂ϕ := {(x, ξ)|hξ, y − xi + ϕ(x) 6 ϕ(y) ∀y ∈ R^d}. (1.1.11) It should be noticed that ∂ϕ is in general a multi-valued operator, i.e. there may be 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 is obtained 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 in one-to-one correspondence with all hyperplanes in R^d=1 that touch the graph of ϕ from ‘below’ (infinitesimally near x even when ϕ is not convex). In this more general context, the equation (1.1.1) is replaced by the following abstract Cauchy problem:

d

dtx(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 consider functions ϕ : R^d → (−∞, +∞] (i.e. ϕ may assume values +∞). In such case the gradient flow of the solutions to (1.1.1) is defined on the closure of the proper domain D(ϕ) := {x ∈ R^d|ϕ(x) < ∞} of ϕ.

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Gradient flows in Hilbert spaces

Our next consideration is whether we really need to work in R^d, or can we extend our results to infinite dimensional Hilbert or perhaps Banach spaces. It turns out that the theory of gradient flows on R^d is easily extended to arbitrary Hilbert spaces, while any other Banach space setting does not work in general, and in order to insure existence of the discrete approximate solutions, as well as the proper solutions in continuous time one typically assumes the lower level sets of ϕ to be relatively compact. The reason that there is no general theory of gradient flows in Banach spaces is simply because the geometry of these spaces does not support such constructions, i.e. suitable variational estimates do not hold in such spaces.

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 that for each α ∈ R any α-convex function is β-convex for any β < α. So α-convex functions with α > 0 stay convex even if we subtract a positive multiple (6 α) of x 7→^|x|₂², 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 various partial 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 lower semi–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

J_h:= (I + h∂ϕ)⁻¹ (1.1.14)

is well defined on H as the unique minimizer of the function R^d3 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:= lim

n→+∞

Jt

n

x0 (1.1.16)

exists.

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3. For each x₀∈ D(ϕ) the curve t 7→ S_tx₀ is the unique solution of the abstract Cauchy problem

d

dtStx0∈ −∂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 compact subinterval in (0, +∞). Moreover, if x0 ∈ D(∂ϕ), then the convergence in (1.1.16) is of order _n¹.

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

_dt^dStx0

are non-increasing.

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

A fundamental example: the heat equation on R^d

Let us give an important example of a partial differential equation that can be interpreted as a gradient flow in a suitable Hilbert space. Recall the heat equation on R^d, i.e.

d

dtu(t, x) = ∆u(t, x) =

d

X

j=1

∂_j∂_ju(t, x), (1.1.20)

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

ϕ(u) :=

(R

R^d|∇u|²dx if u ∈ W^1,2(R^d)

+∞ otherwise. (1.1.21)

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

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analysis is that solutions of certain differential equations are minimizers of an appropriate functional defined on some linear space of functions where one expects to find solutions—the partial differential equation in question is interpreted as an operator on this linear space of functions, and if this operator coincides with the gradient (or more generally with the subdifferential) of a functional ϕ, then any locally minimizing function f of ϕ satisfies ∇ϕ(f ) = 0, which thus amounts to f solving the partial differential equation under consideration. For example, local minimizers of the functional ϕ defined in (1.1.21) are solutions of the Laplace equation

∆u =

d

X

j=1

∂_j∂_ju = 0. (1.1.22)

The techniques of the calculus of variations have turned out to be very powerful in theoretical and applied mathematics and physics, as well as in various practical applications 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 suffice for proving the existence and uniqueness of solutions of the associated Cauchy problem. Thus the question reads: are there reasonable operators A ⊂ H × H such that the Cauchy problem

d

dtx(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 defined by D(A) := {x ∈ H| ∃ξ ∈ H such that [x, ξ] ∈ A}. The reader may observe that the operator A is in fact defined as a relation contained in H × H, hence may be multi-valued. In the course of study of various PDE’s, it has been discovered that allowing ‘operators’ to be multi-valued has definite advantages, and an important class of examples are the subdifferentials of convex, lower semi-continuous functionals defined on Hilbert spaces. Therefore, operators are typically defined as relations contained in H × H, or as mappings H → 2^H. It turned out that the natural 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 called maximal monotonicity conditions. Precisely, a subset A ⊂ H × H is a maximal monotone operator (H is a given Hilbert space), if the following two conditions hold

1. A is a monotone operator, i.e. for each [x1, ξ1], [x2, ξ2] ∈ A we have that hx1− x2, ξ₁− ξ2i > 0 (1.1.24)

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2. A is a maximal element in the class of monotone subsets of H × H with respect 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 for each 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 of convex functionals are maximal monotone operators. More generally, one can consider maximal α-monotone operators for α ∈ R, i.e. operators A ⊂ H × H for which the operator A − αI is a maximal monotone operator. Clearly, the subdifferential 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 be used to construct a ‘numerical approximation’ of the solutions of (1.1.23), the limit Stx0=

Jt n

n

x0exists, and the curve 0 6 t 7→ x(t) := S^tx0is the unique solution, for each initial value x₀ ∈ D(A). Furthermore, (St)_t>0 is a semigroup with the same kind of properties as stated in Theorem 1.1.1, except for one structural difference. In the gradient flow case, for each point x₀ ∈ D(ϕ) = D(∂ϕ) (the equality here is not hard to prove), the curve 0 6 t 7→ Stx₀ is Lipschitz on each compact 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 of maximal monotone operators, the semigroup of solutions does not possess such a regularizing effect, and only the paths emanating from points x0 ∈ D(A) are solutions of (1.1.23) with the local Lipschitz property.

A well known example of an equation that can be interpreted as a Cauchy problem 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 R^d. In order to apply the theory of 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(R^d, dx)×L²(R^d, dx), and define the operator A ⊂ H ×H, A(u, v) := −(v, ∆u), D(A) := D(∂ϕ) × L²(R^d, dx), where ϕ denotes the functional defined in (1.1.21). It can be shown that A is a maximal monotone operator and

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one easily sees that (1.1.27) amounts to d

dt(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 maximal α-monotone operators on Hilbert spaces, are various approximation theorems, and in particular the Trotter-Kato product formula. In order to keep the exposition simple, let us consider a Hilbert space H and two convex lower semi–continuous functionals

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 are such, hence according to Theorem 1.1.1, each of the functionals ϕ₁, ϕ₂, and ϕ, induce gradient flow semigroups, which we denote by (S_t¹)_t>0, (S_t²)_t>0, and (St)_t>0, respectively. Denote moreover J_h¹, h > 0, and J_h², h > 0, to be the resolvent operators associated to ϕ1 and ϕ2, respectively. One version of the Trotter-Kato product formula in this context reads:

J²t

nJ¹t n

ⁿ

x −→ Stx, as n → +∞, t > 0, (1.1.30) for each x ∈ D(ϕ). Moreover, this convergence is uniform on each compact time interval. Furthermore, denoting P¹ and P² to be the nearest point projections onto the closed convex subset E1 := D(ϕ1) and E2 := D(ϕ2), respectively, we have the following version of the product formula:

S²t

n◦ P² S¹t

n◦ P¹ⁿ

x −→ S_tx, 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 where some steps in de approximation procedure may be given by the corresponding semigroup and others by the resolvent. Furthermore, product formulas associated to sums of pairs (or more generally of finite sequences) of maximal monotone operators can be proven, too, provided that the sum operator is also maximal monotone.

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

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or even impossible to handle. Let a convex functional ϕ : H → (−∞, +∞] be given and suppose that we need to restrict the evolution of the system to a convex subset 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 + ψ_B. In general, it is very hard to say anything about the evolution of such a system 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 can actually do a very detailed analysis. Furthermore, product formulas can be used to actually construct ‘sums’ of semigroups and show that these ‘sums’ possess certain properties.

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

The first product formula for matrices was proved in 1959 by Trotter. For any two matrices A, B ∈ R^d×d, Trotter showed that

eⁿ¹^Ae¹ⁿ^Bn n→∞

−→ e^(A+B). (1.1.33)

Observe that any matrix C ∈ R^d×d is the infinitesimal generator of the semigroup of solutions of the ODE system

d

dtx(t) = Cx(t), (1.1.34)

whose semigroup of solutions is defined by

S_t^Cx := e^tCx, t > 0, x ∈ R^d. (1.1.35) This observation naturally leads one to reformulate the Trotter product formula (1.1.33) in terms of semigroups (S_t^A)_t>0 and (S_t^B)_t>0 (defined by (1.1.35) choosing C := A and C := B, respectively), which can be formulated in exactly the same way for linear C0-semigroups on Banach spaces. The product formula for C0-semigroups on Banach spaces was proven in 1960’s. In light of the classical development of the theory of non-linear semigroups induced by accreative operators¹, it seems reasonable to attempt proving product formulas for such semigroups. However, the geometry of general Banach spaces is simply too wild for this 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 we remark that in any Hilbert space accretive operators coincide with the monotone operators

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linear spaces are product formulas for flows induced by maximal monotone operators on Hilbert spaces. Proofs of product formulas for flows on Hilbert spaces induced by maximal monotone operators can be found in [17] (see Proposition 4.3 and Proposition 4.4 there). Furthermore, in 1978 Kato and Masuda have published a proof of the product formula for arbitrary finite sequences of convex lower semi-continuous functionals on Hilbert spaces, where the authors apply a variational calculus method. An extension of the general theory of product formulas 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 of generality beyond the seting of linear spaces. It will be proved that the product formulas associated to finite sequences of geodesically convex functionals defined on complete CAT(0) spaces hold. Since closed convex subsets of Hilbert spaces are 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 the Wasserstein distances on spaces of probability measures

During the past decade, the mathematicians community has witnessed a vast development of a new branch of mathematics called the optimal transportation theory. In this new multidisciplinary field various techniques which descend from classical mathematical disciplines such as calculus of variations, metric geometry and probability theory are jointly applied in order to generate new insights.

This interplay of different disciplines has realized a wide range of applications within mathematics (such as the theory of partial differential equations and the theory of gradient flows on metric spaces) as well as beyond mathematics (such as in image processing, urban planning and even in medical science). It would be hard to give an accurate list of all the monographs that have been published on this topic, for many fellow mathematicians have contributed—optimal transportation theory is certainly one of the major developments in the field of mathematics of the first decade of the 21^st century. The most thorough exposition of the optimal transportation theory available at the preset time, is the book [106] written by the recent Fields medal winner C´edric 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 problem has already been posed in 1781 (see [76]), a satisfactory solution has been given only in recent years. Let us now pose the Monge problem. Suppose that we have a pile of sand at some location and at another location a hole, such that the pile of sand precisely fits in the hole. Assume moreover that we need to fill the hole

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with the sand and that we have to make a certain effort in order to move the sand, say from each point in the pile to each point in the hole, per each unit of mass. The question is: what is the minimal effort that we have to make in order to move the sand? A related problem is the following. A producer of certain goods has a number of factories where the goods are produced (say k), and a number of locations (stores) where these goods are sold. The producer needs to transport the goods from his factories to his stores and he must pay a certain price per unit of 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 mathematical formulation is needed. Let metric spaces X and Y be given and let µ and 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 a B(X) × B(Y ) \ B([0, +∞]) measurable (cost) function. The problem reads: minimize the expression

I(r) :=

Z

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 admissible functions, i.e. measurable functions which map X into Y and push the probability measure 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 = R^d (for some d ∈ N), then ν = r^#µ means that the law of r under µ is ν. Obviously, for each admissible function r : X → Y , the expression I(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 admissible function which realizes the infimum in (1.2.1), and such minimizer functions are said to be solutions of the Monge problem associated to X, µ, Y, ν and the cost function c. Some basic examples of a Monge problem are obtained by taking X = Y = R^d, c(x, y) := |x − y|^p, for all x, y ∈ R^d, with p ∈ [1, +∞). A case of particular relevance for applications in PDE’s and gradient flows theory is where p = 2. One may observe that the examples of the sand pile and the transportation of goods present rather specific situations, from the point of view of the general formulation. In the sand pile example, both metric spaces X (the location of the sand pile) and Y (the location of the hole) are in fact subsets of the 3-dimensional Euclidean space, and both probability measures µ and ν are in fact suggested to be uniform measures on X and Y —sand should have more or less the same density everywhere in the pile and hole. In the example of the transport of goods, both spaces X (the factories) and Y (the stores) are metric spaces consisting of finitely many points.

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

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

f (x) = g(r(x))| det Dr(x)|, (1.2.3) where Dr denotes the Jacobian of r. Clearly, finding any admissible functions is not 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. The relaxed problem is in fact a linear problem on a space of measures and it is much easier to solve. Given separable metric spaces X and Y , probability measures µ and 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 the following functional

T (σ) :=

Z

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 by πX(x, y) = x, πY(x, y) = y for x ∈ X, y ∈ Y . Thus a minimizer is a probability measure on B(X) × B(Y ), and any such minimizer is said to be a solution of the Kantorovich problem associated to the data X, Y, µ, ν, c.

The Kantorovich problem is a relaxed version of the Monge problem in the following sense. Suppose that r : X → Y is an admissible function for the Monge problem. Defining (iX, r) : X → X × Y , (iX, r)(x) := (x, r(x)), one easily sees that 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 speaking the Kantorovich problem is a relaxed version of the Monge problem, since the Kantorovich formulation allows splitting of the mass (from the departure space X), contrary to the Monge formulation. It is now easily seen that given any data X, Y, µ, ν, c, the infimum in the Kantorovich problem is less or equal the infimum in the Monge problem—in particular a solution of the Monge problem (with given data X, Y, µ, ν, c) may not generate a solution of the Kantorovich problem. The reader may now indeed observe that the Kantorovich problem is a linear problem in the linear space of measures, or rather its convex subset of probability measures on B(X) × B(Y ). Kantorovich won the Nobel price in 1940’s for his related work in the field of economy.

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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), and c(x, y) := d^p(x, y), x, y ∈ X, with p ∈ [0, +∞). In this case, the Kantorovich problem always has solutions (by means of weak compactness and convexity argu- ments), and the set of solutions is called the set of optimal admissible transportation plans, denoted Γo(µ, µ) ⊂ P(X × X) (in this notation the exponent p is not accounted). Furthermore, the set of all admissible transportation plans is usually denoted 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), for any separable metric space X. These distances are (to some extent erroneously) called the Wasserstein p distances, denoted Wp, and they are given by choosing X = Y and c(x, y) := d^p(x, y) in the Kantorovich problem (d denotes the distance function on our separable metric space X, and p ∈ [1, +∞)), and taking the p-th root of the infimum of the functional T over all σ ∈ Γ(µ, ν). This infimum is then a minimum, but the set of minimizers is in general not a one point set. As a matter a fact, one easily sees that the set of minimizers is a convex subset of P(X) (with respect to the usual linear structure of the set of Borel measures on X), hence one either has one minimizer, or infinitely many.

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

Z

X

d^p(x, y) dµ(x) < +∞. (1.2.6) Alternatively, one may choose to work with a pseudo-distance, i.e. allow that the distance function assumes the value +∞. Such an approach has turned out useful if one considers the Wasserstein distances on the set probability measures on the Wiener space (see [41], for some fundamental pioneering work on this topic).

Wasserstein spaces and in particular the quadratic case p = 2 have turned out to be very useful in studying the geometry of the space X itself. They have also found 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 the existence of solutions of the Monge problem in Euclidean spaces and on manifolds, respectively, while in [41] a fundamental pioneering work on the Monge problem in the infinite dimensional setting can be found. The list of works where the optimal transportation theory and the Wasserstein distances are applied to treat PDE’s would probably be too long to fit in this thesis. However, the following works are considered to be milestones in this genesis: [16], [54], [85], [22], [5], [21]. Striking results 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 curvature bounded from below by some K ∈ R. These works are inspired by the fact that on a Riemannian manifold the lower Ricci curvature is bounded by K if and only

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

1.3 Gradient flows in metric spaces

The classical theory of gradient flows on Hilbert spaces, which is briefly sketched in Section 1.1, has been extended in the monograph [5] to a more general setting of complete metric spaces which satisfy an appropriate geometric type of assumption.

In this section we present a concise overview of these results. The earliest known results regarding formulating the gradient flow equations in a purely metric setting, as well as some fundamental notions presently used originate from the Italian mathematician De Giorgi². De Giorgi was lead to consider this problem through his work on a PDE in L²(R^d) where the system under consideration was restricted to the unit sphere. Since the unit sphere in L²(R^d) is a non-convex subset the available 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 the remainder 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 ∈ AC^p([a, b], X), if there is a non-negative function v ∈ L^p([a, b]) such that for a 6 s 6 t 6 b we have that

d(γ(s), γ(t)) 6 Z t

s

v(r)dr. (1.3.1)

This definition is inspired by the classical result of analysis which asserts that a function 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 an absolutely 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 AC^p([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 ∈ L^p([a, b]) (in the Lebesgue a.e sense) that satisfies (1.3.1) (see [5] Theorem 1.1.2.).

Notice that if X is a Hilbert space, and γ ∈ AC^p([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 the norm of the derivative ˙γ(t) = lim_h→0γ(t+h)−γ(t)

h .

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

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Metric slope of a functional

Another important concept in the context of gradient flows on metric spaces is the metric slope of a functional defined on X. As in Hilbert spaces, a functional defined on a metric space X is by definition a mapping ϕ : X → (−∞, +∞] = R ∪ {+∞}.

The metric slope denoted |∂ϕ| associated to a functional ϕ : X → (−∞, +∞], is defined by

|∂ϕ|(x) := lim sup

y→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. Now observe 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 metric slope of a functional: in a general metric space X, one cannot define −∇ϕ as the 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 Variational Inequality (EVI)

In order to provide a theory of gradient flows in any metric space setting (where no linear or even convex structure in the classical sense is available), one must reformulate the abstract Cauchy problem

˙

x(t) ∈ −∂ϕ(x(t)), L¹ a.e. t > 0, x(0) = x0∈ D(∂ϕ)(= D(ϕ)), (1.3.4) (L¹denotes the restriction of one-dimensional Lebesgue measure to [0, +∞)) in the first place. The objects that we have at hand to give any alternative formulation of the abstract Cauchy problem, are the functional ϕ, the distance function d, the metric slope |∂ϕ|, and the metric derivative of absolutely continuous curves.

In order to give an alternative purely metric formulation of the evolution problem (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

hh, z − xi 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

hh, z − xi + 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 for each z ∈ D(ϕ)

h−x(t), z − x(t)i +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)

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Observe moreover that _dt^d¹₂|x(t) − z|² = hx(t), x(t) − zi 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 d

dt 1

2d²(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. The inequality 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 that it can be posed in any metric space. Indeed, given a metric space X, a functional ϕ : X → (−∞, +∞], and a point x0∈ D(ϕ), one can also pose the problem of the existence of a curve 0 6 t 7→ x(t) ∈ X, which is continuous on [0, +∞), starts at x₀, 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 known that if X is a Banach space but not a Hilbert space, even if ϕ is convex and lower semi-continuous, there may be no solution to this problem, while in Hilbert spaces solutions 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 of the solution curve, and such formulation is also suitable to pose in general metric spaces. We will give this characterisation below, but we remark that it is not used elsewhere 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) := _dt^dx(t) = −∇ϕ(x(t)) for t > 0 implies that d

dtϕ(x(t)) = h∇ϕ(x(t)), ˙x(t)i = −|∇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) are negative 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 d

dtϕ(x(t)) = −1

2| ˙x(t)|²−1

2| − ∇ϕ(x(t))|², t > 0. (1.3.11)

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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 h∇ϕ(x(t)), ˙x(t)i = d

dtϕ(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 ¹₂(| ˙x|(t) − |∇ϕ(x(t))|)²6 0 for t > 0, then holds too. Furthermore, as we always have that

d

dtϕ(x(t)) = h∇ϕ(x(t)), x(t)i > −| − ∇ϕ(x(t))|| ˙x(t)| >

> −1

2| − ∇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 a metric space (X, d) and a functional ϕ : X → (−∞, +∞], find a continuous curve x : [0, ∞) → X, which starts at x0∈ X of class AC²([a, b], X), such that

d

dtϕ(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 no assertion 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 not to be a good setting for generalising gradient flow theory without further assumptions, such as compactness of the level sets of the functional under consideration), some geometric properties of the underlying space X should clearly play the key role 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 by definition 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 a geodesic. 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.

(32)

1 one has that d(γ(t), γ(s)) = |t−s|d(γ(0), γ(1)), and this is the standard definition of geodesics (or geodesic curves) in metric spaces. Hence a natural definition of the convexity of a functional ϕ defined on a metric space X reads: for each geodesic γ : [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 inequality as (1.3.6) for all geodesics γ in X, but with the term^α₂t(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 of points 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 distances between points laying on geodesic triangles with distances between points on the associated comparison triangles. More precisely, if X is a metric space, x, y, z ∈ X, and γ1, γ2and γ3are 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 particular choice of such a triple of points in R² is immaterial since any two such a triples are 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 defined by 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 x0with x1the following holds:

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 R²is a Hilbert space, the property 2. above equivalently states that for all v, x0, x1 ∈ X, a given geodesic γ that joins x0 with x1, and for each associated comparison triangle ¯v, ¯x0, ¯x1 ∈ R², denoting ¯γ(t) := (1 − t)¯x0+ t¯x1, 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 if and only if it is simply connected and has globally non-positive sectional curvature.

Geometric approach to evolution problems in metric spaces

Geometric approach to evolution problems in metric spaces

Proefschrift

ter verkrijging van

de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus

prof. mr. P.F. van der Heijden,

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

klokke 15:00 uur

door Igor Stojkovi´ c geboren te Belgrado

in 1972

promotor: prof.dr. S.M. Verduyn Lunel copromotor: dr.ir. O.W. van Gaans

overige leden: prof.dr. L. Ambrosio (Scuola Normale Superiore, Pisa, Itali¨ e)

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

Contents

Acknowledgment

Samenvatting

Chapter 1

Introduction

1.1 Gradient flows, maximal monotone operators and product formulas in Hilbert spaces

1.1.1 Gradient flows

1.1.2 Maximal monotone operators

1.1.3 Trotter-Kato product formulas

1.2 The Monge-Kantorovich problem and the Wasserstein distances on spaces of probability measures

1.2.1 The Monge problem

1.2.2 The Kantorovich problem

1.2.3 Wasserstein distances

1.3 Gradient flows in metric spaces