Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Bowles, Malcolm
Doctor of Philosophy - PhD
University of British Columbia
In this work, we introduce and study a class of convex functionals on pairs of probability measures, the linear transfers, which have a structure that com- monly arises in the dual formulations of many well-studied variational prob- lems. We show that examples of linear transfers include a large number of well-known transport problems, including the weak, stochastic, martingale, and cost-minimising transports. Further examples include the balayage of measures, and ergodic optimisation of expanding dynamical systems, among others. We also introduce an extension of the linear transfers, the convex transfers, and show that they include the relative entropy functional and p-powers (p ≥ 1) of linear transfers. We study the properties of linear and convex transfers and show that the inf-convolution operation preserves their structure. This allows dual formu- lations of transport-entropy and other related inequalities, to be computed in a systematic fashion. Motivated by connections of optimal transport to the theory of Aubry- Mather and weak KAM for Hamiltonian systems, we develop an analog in the setting of linear transfers. We prove the existence of an idempotent operator which maps into the set of weak KAM solutions, an idempotent linear transfer that plays the role of the Peierls barrier, and we identify analogous objects in this setting such as the Mather measures and the Aubry set. We apply this to the framework of ergodic optimisation in the holonomic case.
Linear transfers, Kantorovich operators, and theirergodic propertiesbyMalcolm BowlesB. Sc. University of Victoria, 2012M. Sc, University of Victoria, 2014a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Mathematics)The University of British Columbia(Vancouver)August 2020c© Malcolm Bowles, 2020The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Linear transfers, Kantorovich operators, and their ergodicpropertiessubmitted by Malcolm Bowles in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics.Examining Committee:Nassif Ghoussoub, MathematicsSupervisorYoung-Heon Kim, MathematicsSupervisorPhilip Loewen, MathematicsSupervisory Committee MemberStephen Gustafson, MathematicsUniversity ExaminerGordon Semenoff, PhysicsUniversity ExamineriiAbstractIn this work, we introduce and study a class of convex functionals on pairs ofprobability measures, the linear transfers, which have a structure that com-monly arises in the dual formulations of many well-studied variational prob-lems. We show that examples of linear transfers include a large number ofwell-known transport problems, including the weak, stochastic, martingale,and cost-minimising transports. Further examples include the balayage ofmeasures, and ergodic optimisation of expanding dynamical systems, amongothers. We also introduce an extension of the linear transfers, the convextransfers, and show that they include the relative entropy functional andp-powers (p ≥ 1) of linear transfers.We study the properties of linear and convex transfers and show that theinf-convolution operation preserves their structure. This allows dual formu-lations of transport-entropy and other related inequalities, to be computedin a systematic fashion.Motivated by connections of optimal transport to the theory of Aubry-Mather and weak KAM for Hamiltonian systems, we develop an analog inthe setting of linear transfers. We prove the existence of an idempotentoperator which maps into the set of weak KAM solutions, an idempotentlinear transfer that plays the role of the Peierls barrier, and we identifyanalogous objects in this setting such as the Mather measures and the Aubryset. We apply this to the framework of ergodic optimisation in the holonomiccase.iiiLay SummaryA probability measure specifies the chance of a particular event occurring.Many mathematical problems are concerned with computing similarities/d-ifferences between two probability measures. Among such problems, a largenumber of them have a common structure; we isolate and define this com-mon formalism as a “linear transfer”, and more generally, a “convex trans-fer”. We demonstrate linear and convex transfers encompass a wide rangeof well-known mathematical problems and study their properties.A Hamiltonian system is a system whose state over time is determinedby a function describing its total energy. In trying to investigate the long-time dynamics of this system, one is led to the search for functions whichsolve certain equations involving certain constants; this is collectively knownas Aubry-Mather/weak KAM theory. We generalise aspects of this theoryto the setting of linear transfers.ivPrefaceThis thesis is based on [10], which is currently in preparation to be submittedfor publication.The material presented in Chapters 2 and 3 is based on initial ideas ofNassif Ghoussoub, which we subsequently developed and expanded jointly.The material in Chapter 4 is an overview of classical Aubry-Mather theory,which motivated the present work, and was written by myself. The mate-rial presented in Chapter 5 is work developed in collaboration with NassifGhoussoub. Section 5.6.1 is based on a key contribution by Dorian Martino,which I modified and extended.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Linear and convex transfers . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The Legendre-Fenchel transform and duality for convex func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Linear transfers . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Kantorovich operators . . . . . . . . . . . . . . . . . . . . . . 122.5 First examples of linear transfers . . . . . . . . . . . . . . . . 182.5.1 Convex energies . . . . . . . . . . . . . . . . . . . . . 192.5.2 Linear transfers induced from positively homogenousoperators . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.3 Balayage of measures . . . . . . . . . . . . . . . . . . 242.6 Linear transfers and optimal mass transportation . . . . . . . 282.7 Representation and envelopes of linear transfers . . . . . . . . 33vi2.7.1 Linear transfers and optimal weak transports . . . . . 332.7.2 Linear transfer envelopes . . . . . . . . . . . . . . . . 372.7.3 Recessions of linear transfers . . . . . . . . . . . . . . 412.8 Linear transfers which are not mass transports . . . . . . . . 442.8.1 Linear transfers associated to weak mass transports . 442.8.2 One-sided transfers associated to stochastic mass trans-port . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.8.3 Transfers associated to optimally stopped stochastictransports . . . . . . . . . . . . . . . . . . . . . . . . . 492.9 Convex transfers . . . . . . . . . . . . . . . . . . . . . . . . . 512.9.1 Backward entropic transfers . . . . . . . . . . . . . . . 592.10 Operations on convex and linear transfers . . . . . . . . . . . 592.10.1 Linear transfers are a convex cone . . . . . . . . . . . 592.10.2 Inf-convolution . . . . . . . . . . . . . . . . . . . . . . 602.10.3 Tensor products and dual sums of linear transfers . . . 622.10.4 Projections and Hopf-Lax formulae . . . . . . . . . . . 642.11 Subdifferentials of convex transfers . . . . . . . . . . . . . . . 693 Dualities for transfer inequalities . . . . . . . . . . . . . . . 723.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.2 Backward convex coupling and backward linear transfer . . . 753.3 Forward convex coupling and backward linear transfer . . . . 783.3.1 Moment measures . . . . . . . . . . . . . . . . . . . . 803.4 Maurey-type inequalities . . . . . . . . . . . . . . . . . . . . . 824 Hamiltonian/Lagrangian dynamics: Weak KAM, Aubry-Mather, and optimal transport . . . . . . . . . . . . . . . . . 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . 864.3 Aubry-Mather theory . . . . . . . . . . . . . . . . . . . . . . 914.3.1 Man˜e´ critical value . . . . . . . . . . . . . . . . . . . . 914.3.2 Peierls Barrier and the Aubry set . . . . . . . . . . . . 934.3.3 Mather measures . . . . . . . . . . . . . . . . . . . . . 96vii4.4 Weak KAM theory . . . . . . . . . . . . . . . . . . . . . . . . 1004.5 Connections of weak KAM and Aubry-Mather to optimaltransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015 Weak KAM and Aubry-Mather for linear transfers . . . . 1065.1 Extension of Kantorovich operators from C(Y ) to USCσ(Y ) . 1075.1.1 Conjugate functions for forward and backward lineartransfers . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2 Man˜e´ constant and weak KAM solutions for backward lineartransfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3 Idempotent linear transfers . . . . . . . . . . . . . . . . . . . 1215.3.1 Examples of idempotent linear transfers . . . . . . . . 1255.3.2 T -Lipschitz functionals . . . . . . . . . . . . . . . . . 1285.4 Ergodic properties of equicontinuous semigroups of transfers . 1325.4.1 The case of optimal transport for continuous cost func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.4.2 Aubry-Mather theory and weak KAM theory for La-grangian systems . . . . . . . . . . . . . . . . . . . . . 1435.4.3 The Schro¨dinger semigroup . . . . . . . . . . . . . . . 1465.5 Regularization and ergodic properties of non-continuous lin-ear transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.5.1 Regularization . . . . . . . . . . . . . . . . . . . . . . 1475.5.2 Ergodic properties for non-continuous linear transfersvia regularisation . . . . . . . . . . . . . . . . . . . . . 1515.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1575.5.4 Interpolation and ergodic properties . . . . . . . . . . 1615.6 Ergodic properties for non-continuous linear transfers withcontrol on convergence to c(T ) . . . . . . . . . . . . . . . . . 1645.6.1 Linear transfers and ergodic optimization . . . . . . . 1725.6.2 Ergodic optimization in the deterministic holonomicsetting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.6.3 Ergodic optimization in the stochastic holonomic setting184viiiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191ixAcknowledgmentsFirst, I would like to thank Nassif Ghoussoub for this work would not havebeen possible without his support and collaboration. I also want to expressmy gratitude to Young-Heon Kim for his valuable advice, guidance, andgenerosity, throughout my time at UBC.I would also like to thank Philip Loewen for his careful reading andvaluable suggestions to improve the thesis, along with Stephen Gustafsonand Gordon Semenoff for their time as the university examiners.A special thanks to all the people I met at UBC, especially my friends andfellow graduate students, Sarafa Iyaniwura, Kelsey Everard, and VirginieSonnet, who have substantially enriched my time here and am so happy tohave met.I especially appreciate all the discussions, both mathematical, and non-mathematical, with my fellow graduate students Guillermo Martinez-Dibene,Sophie Macdonald, Alperen Bulut, Igor Pinheiro, Li Wang, and Reza Sadoughian.I wish them success in all their endeavours. I also would like to thank AaronPalmer, Samer Dweik, and Hugo Lavenant, for their helpful discussions.Finally, I save until the end, my thanks to my Mum, who has alwaysbeen a constant source of encouragement, and my sisters Alison and Rachel.I would not be writing this today without their support.xChapter 1IntroductionThis work is devoted to an axiomatic study of certain convex functionalson pairs of probability measures, whose structure commonly arises in thedual formulations of many well-studied variational problems. Typical ex-amples include, the weak, stochastic, martingale, and cost-minimising trans-port problems, the relative entropy, the balayage of measures, and ergodicoptimisation of expanding dynamical systems, among others.The overall broad aim is to unify the settings of these various problemstogether, with the goal to add clarity and understanding, and to derive non-trivial extensions, including in this work, an analog of the Aubry-Matherand weak KAM theory for Lagrangian systems. We begin by introducing inChapter 2 the notions of backward/forward linear couplings, as those convexfunctionals on pairs of probability measures (µ, ν) which arise as the supre-mum of linear functionals (µ, ν) 7→ ∫Y gdν − ∫X fdµ where the supremum istaken over pairs of continuous functions (g, f) which lie on the graph of anoperator ; either (g, T−g), or (T+f, f) for operators T−, T+. Among thesebackward/forward linear couplings, there exists a distinguished subset thatwe call linear transfers. These possess an additional structure that arisesas a consequence of being defined on pairs of measures (µ, ν) and there-fore allow us to consider one-variable convex maps by fixing µ, or fixing ν.A backward/forward linear transfer is a backward/forward linear couplingfor which, when fixing µ or ν, and considering the resulting convex func-1tion in ν or µ, its Legendre transform precisely coincides with the functiong 7→ ∫X T−gdµ or f 7→ ∫Y −T+(−f)dν. In other words, the operator T− orT+ completely characterises the Legendre duality. This leads to the studyof such operators T−, T+, which we call backward/forward Kantorovich op-erators who, conversely, define a linear transfer. The Kantorovich operatorsare the main focus for an “Aubry-Mather and weak KAM theory for lineartransfers” later in Chapter 5.We proceed by exhibiting a number of examples in Section 2.5 which canbe realised as backward/forward linear transfers. These include all convexenergies ν 7→ I(ν) of one variable, and any transfer whose Kantorovich oper-ator is given by a point transformation σ : X → X on the underlying spaceX, or more generally, by a positive bounded linear operator T on continuousfunctions; this is connected to ergodic theory for expanding dynamical sys-tems which we discuss later in Chapter 5. Interesting examples also includethe balayage of measures, which is concerned with pairs of measures (µ, ν)that are in partial order with respect to convex cones A of continuous func-tions, of which lies the work of Strassen, the important theory developedby Choquet for convex functions when A is the cone of convex functions,and the work of Skorokhod for Brownian motion when the cone is subhar-monic functions. In Section 2.6, we show that cost-minimising transport isa linear transfer. This is especially important for us as a connection madeby Bernard-Buffoni between cost-minimising transport, and Aubry-Matherand weak KAM theory, provides an inspiration for an analog in the settingof linear transfers in Chapter 5. In Section 2.7, we show that, under certainassumptions, linear transfers actually coincide with weak transports studiedby Gozlan et. al. and permits us to introduce the notion of a “linear transferenvelope” in an analogy to classical convex envelopes. In Section 2.8, we in-troduce further examples of linear transfers, including martingale transport,the Schro¨dinger bridge, stochastic transport, and related problems.We next introduce a natural extension of linear transfers in Section 2.9,the convex transfers. As with linear couplings and linear transfers, we de-fine convex couplings as those functionals T (µ, ν) which are supremums oflinear couplings, and convex transfers as the distinguished subset of convex2couplings for which, when restricting to the one variable convex functionν 7→ T (µ, ν) (similarly for µ 7→ T (µ, ν)), their Legendre transform is givenby g 7→ infi∈I∫X T−i gdµ, an infimum over a family of operators T−i . An im-portant example of a convex, but not linear, transfer is the relative entropy.We then introduce operations that preserve linear and convex transfers inSection 2.10, the most important of which is the notion of inf-convolutionwhich we shall extensively use in Chapter 5. Finally, considered as onevariable convex functions ν 7→ T (µ, ν) or µ 7→ T (µ, ν), we discuss theirsubdifferentials in Section 2.11.In Chapter 3, we use the structure of linear and convex transfers towrite equivalent statements for “transport-entropy” type inequalities andvarious generalisations. A typical transport-entropy inequality is of theform Tc(ν, µ) ≤ H(µ, ν) where Tc(µ, ν) is an optimal transport for somechosen cost function c between ν and µ, and H(µ, ν) is the relative en-tropy of ν with respect to µ. Here µ is a fixed reference measure, and theinequality should hold for all ν. The inequality is equivalent to the pos-itivity of infν∈P(Y ){H(µ, ν) − Tc(ν, µ)}. By using the structure of linearand convex transfers, we can write in a systematic way a dual formula forinfν∈P(Y ){H(µ, ν)− Tc(ν, µ)}, as well as various other types of inequalities,for which the positivity could be obtained.Chapter 4 is a preliminary to Chapter 5, which consists of an overview ofthe theory of weak KAM developed by Fathi. In Section 4.2, we begin by in-troducing integrable Hamiltonian systems and the role the Hamilton-Jacobiequation plays in the search for a change of coordinates that transform anon-integrable Hamiltonian to an integrable one. We then proceed in Sec-tion 4.3 to introduce the Peierls barrier from which we may construct theimportant Aubry set where viscosity subsolutions of Hamilton-Jacobi aredifferentiable, and on which the Mather measures are supported. The Lax-Oleinik semi-group of operators and weak KAM solutions of Fathi are dis-cussed in Section 4.4 which connects solutions of Hamilton-Jacobi with theAubry set and Mather measures. An important characterisation establishedby Bernard-Buffoni is then presented in Section 4.5, where they characterisethe Peierls barrier, Mather measures, Aubry set, and weak KAM solutions,3via optimal transport.In Chapter 5, which consists of the main application in this work, weare interested in developing an analog of the weak KAM and Aubry-Mathertheory of Chapter 4, for linear transfers. Here the Kantorovich operatorsT−, T+, introduced in Chapter 2, play the role of the Lax-Oleinik semi-group of the Chapter 4 in defining weak KAM solutions for linear transfers.These are the functions f that satisfy T+f − c = f , or those g satisfyingT−g + c = g for a specific (critical) constant c. Through Theorems 5.4.3,5.5.3, and 5.6.3, in Sections 5.4, 5.5, and 5.6, we show that for most lineartransfers, there exists an idempotent Kantorovich operator T∞ which mapsinto the set of weak KAM solutions.Theorem 1.0.1 (Theorem 5.4.3). Let T be a weak∗ continuous backwardlinear transfer on M(X) ×M(X) with modulus of continuity δ, and withbackward Kantorovich operator T− : C(X)→ C(X). Then c(T ) := infµ T (µ, µ)is the unique constant for which there exists a backward Kantorovich opera-tor T−∞ : C(X) → C(X), together with its induced backward linear transferT∞, satisfying:1. T−∞ maps every g ∈ C(X) to a backward weak KAM solution for T−,i.e.,T−n ◦ T−∞g + nc(T ) = T−∞g for all g ∈ C(X) and all n ∈ N.2. T−∞ is idempotent, i.e. T−∞ ◦ T−∞g = T−∞g for all g ∈ C(X).3. T∞ satisfies,(Tn−nc(T ))?T∞(µ, ν) = T∞(µ, ν) = T∞?(Tn−nc(T ))(µ, ν) for every n ∈ N.4. T∞ is idempotent and therefore A-factorisable, i.e. the set A := {σ ∈P(X); T∞(σ, σ) = 0} is non-empty, and for every µ, ν ∈ P(X), wehaveT∞(µ, ν) = inf{T∞(µ, σ) + T∞(σ, ν), σ ∈ A},and the infimum on A is attained.45. For every µ, ν ∈ P(X), we havesup{∫T−∞gd(ν − µ) ; g ∈ C(X)}≤ T∞(µ, ν) ≤ lim infn→∞ (Tn(µ, ν)−nc(T )).6. If T (µ, µ) = c(T ), then µ ∈ A. Additionally, (µ, µ) ∈ D if and only ifµ ∈ A and T (µ, µ) = c(T ), whereD := {(µ, ν) ∈ P(X)× P(X) : T (µ, ν) + T∞(ν, µ) = c(T )}.Inspired by the optimal transport analysis provided by Bernard-Buffoniin the previous chapter, we can identify analogs of the objects from Aubry-Mather and weak KAM. The Peierls barrier in this setting is the idempotentlinear transfer associated to the Kantorovich operator T−∞. The Mathermeasures consist of those minimising transport plans for the weak transportassociated to the linear transfer T , and the Aubry set is the set D.Moving from weak∗ continuity to weak∗ lower semi-continuity, which isthe natural setting for linear transfers, yields the following.Theorem 1.0.2 (Theorem 5.5.3). Let T be a backward linear transfer suchthat D1(T ) contains the Dirac measures. Assume c(T ) := infµ T (µ, µ) <+∞, and that inf(µ,ν) T (µ, ν) = infµ T (µ, µ).Then c(T ) is the unique constant for which there exists an idempotentKantorovich operator T−∞ : C(X)→ USC(X) mapping into the set of back-ward weak KAM solutions, i.e.T−n ◦ T−∞g + nc(T ) = T−∞g for all g ∈ C(X) and all n ∈ N.The corresponding backward linear transfer T∞ is idempotent, the set A :={σ ∈ P(X); T∞(σ, σ) = 0} is non-empty, and for every (µ, ν), satisfiesT∞(µ, ν) = inf{T∞(µ, σ) + T∞(σ, ν), σ ∈ A},and the infimum on A is attained.Finally, the existence of an idempotent Kantorovich operator mapping5into the set of backward weak KAM solutions breaks down when we weakenthe hypotheses on the linear transfer.Theorem 1.0.3 (Theorem 5.6.3). Suppose T is a backward linear transferon P(X)× P(X) such that D1(T ) contains all Dirac measures. Assume:1. c(T ) := infµ T (µ, µ) < +∞,2. supx∈X infν∈P(X) T (δx, ν) < +∞,3. there exists K > 0 such thatlim supn→∞{nc(T )− infµ,ν∈P(X)Tn(µ, ν)} ≤ K,4. there exists g ∈ C(X), such that the functionx 7→ lim infn→∞ (T−n g(x) + nc(T ))belongs to USCb(X).Then there exists h ∈ USC(X) such that T−h+ c(T ) = h on X.In the last Section 5.6.1, we apply our results to ergodic optimization,in particular, to the setting of symbolic dynamics.6Chapter 2Linear and convex transfers2.1 IntroductionThroughout this chapter and entire thesis, X shall denote a compact metricspace, equipped with its Borel σ-algebra. Where necessary, we will write themetric on X by dX . The collection of Borel signed measures will be denotedby M(X), which is equipped with the weak∗ topology in duality with thereal-valued continuous functions C(X); the subset of Borel probability mea-sures on X will be written as P(X). The set of functions f : X → R∪{−∞}which are upper semi-continuous shall be denoted USC(X); the subset ofbounded functions in USC(X) shall be denoted by USCb(X). The notationLSC(X) is the set of lower semi-continuous functions, i.e. those functionsf for which −f ∈ USC(X); similarly for LSCb(X). The set of functionsf : X → R ∪ {±∞} which are Borel-measurable will be denoted by B(X);the subset of B(X) which are bounded above is Bb(X); those which arebounded below is Bb(X). The same conventions above hold for a secondspace Y in place of X.Throughout, T : P(X)×P(Y )→ R∪{+∞} will denote a proper (i.e. notidentically +∞), bounded below, convex, and weak∗ lower semi-continuousfunctional. By convex, we will always intend to mean joint convexity :7If µ0, µ1 ∈ P(X), and ν0, ν1 ∈ P(Y ), thenT (µλ, νλ) ≤ (1− λ)T (µ0, ν0) + λT (µ1, ν1), for all λ ∈ [0, 1],where µλ := (1− λ)µ0 + λµ1 and νλ := (1− λ)ν0 + λν1.To make precise the weak∗ lower semi-continuity of T , we note thatthe weak∗ topology on P(X) is metrizable when X is compact, and themetric can be taken to be the Wasserstein distance of optimal transportwith distance cost given by the metric dX on X (see e.g. [60], Chapter 7,and also Section 2.6 of this thesis for more on optimal transport). Therefore,weak∗ lower semi-continuity is equivalent in this case to sequential weak∗lower semi-continuity, which we recall is the following property:If (µn) ⊂ P(X), µ∞ ∈ P(X), such that µn → µ∞ as n → ∞ in theweak∗ sense (and similarly for a sequence (νn) ⊂ P(Y ), and ν∞ ∈ P(Y )),thenT (µ∞, ν∞) ≤ lim infn→∞ T (µn, νn).Throughout, T shall also be viewed implicitly as a function on M(X) ×M(Y ) by defining T ≡ +∞ on M(X) ×M(Y )\P(X) × P(Y ). Its firstpartial domain D1(T ) is the set of all µ ∈ P(X) such that T (µ, ν) < +∞for some ν ∈ P(Y ); similarly its second partial domain is D2(T ) the set ofall ν ∈ P(Y ) such that T (µ, ν) < +∞ for some µ ∈ P(X).2.2 The Legendre-Fenchel transform and dualityfor convex functionsWe briefly recall the Legendre-Fenchel transform and the Fenchel-Moreau-Rockafellar theorem for convex functions (see e.g. [2], Section 9.3).Definition 2.2.1. Let V be a normed linear space and f : V → R∪ {+∞}a proper function. The Legendre-Fenchel transform (or conjugate) off is the function f∗ : V ∗ → R∪{+∞} defined on the topological dual spaceV ∗ viaf∗(v∗) := sup{〈v∗, v〉 − f(v) ; v ∈ V }The collection of convex functions f on V which are proper and lower8semi-continuous are distinguished by the fact that f can be recovered fromf∗; this is the following Fenchel-Moreau theorem.Theorem 2.2.2. Let V be a normed linear space and V ∗ its topological dualspace, equipped with the weak∗ topology. Then the following hold:1. If f : V → R ∪ {+∞} is a proper, convex, and lower semi-continuousfunction, thenf(v) = sup{〈v∗, v〉 − f∗(v∗) ; v∗ ∈ V ∗}where f∗(v∗) := sup{〈v∗, v〉 − f(v) ; v ∈ V }.2. If g : V ∗ → R ∪ {+∞} is a proper, convex, and weak∗ lower semi-continuous function, theng(v∗) = sup{〈v∗, v〉 − g∗(v) ; v ∈ V }.where g∗(v) := sup{〈v∗, v〉 − g(v∗) ; v∗ ∈ V ∗} for v ∈ V .2.3 Linear transfersWhen the normed linear space V of the last section is the space of continuousfunctions C(Y ) equipped with the supremum norm, then Riesz’ theoremsays that V ∗ can be identified withM(Y ). Therefore according to Fenchel-Moreau, any functional F : M(Y ) → R ∪ {+∞} which is proper, convex,and weak∗ lower semi-continuous, satisfiesF (ν) = supg∈C(Y ){∫Ygdν − F ∗(g)} (2.1)where F ∗(g) = supν∈M(Y ){∫Y gdν − F (ν)}.For a proper, convex, bounded below, and weak∗ lower semi-continuousfunctional T on P(X) × P(Y ), viewed as a function on M(X) × M(Y )(recall the notation in Section 2.1), we may then consider the partial map,Tµ : ν 7→ T (µ, ν)9for µ ∈ D1(T ). This is a proper, convex, bounded below, and weak∗ lowersemi-continuous function on M(Y ), hence identifying F = Tµ in (2.1)T (µ, ν) = Tµ(ν) = supg∈C(Y ){∫Ygdν − T ∗µ (g)}, µ ∈ D1(T ), ν ∈ P(Y ). (2.2)In exactly the same way, the other partial map Tν : µ 7→ T (µ, ν) on M(X)satisfiesT (µ, ν) = Tν(µ) = supf∈C(X){∫Xfdµ− T ∗ν (f)}, µ ∈ P(X), ν ∈ D2(T ).(2.3)The linear transfers are then those T whose partial maps Tµ, Tν haveLegendre-Fenchel transforms which are linear with respect to µ and ν, re-spectively.Definition 2.3.1 (Linear transfers). Let T : P(X)×P(Y )→ R∪{+∞} bea proper, convex, bounded below, and weak∗ lower semi-continuous function.1. We say that T is a backward linear transfer, if there exists anoperator T− : C(Y ) → F b(X) such that for all µ ∈ D1(T ) and allg ∈ C(Y ),T ∗µ (g) =∫XT−gdµ. (2.4)2. We say T is a forward linear transfer, if there exists an operatorT+ : C(X)→ Fb(Y ), such that for all ν ∈ D2(T ) and all f ∈ C(X),T ∗ν (f) =∫Y−T+(−f)dν.Remark 2.3.2. It is important to stress that the operator T− (resp., T+)is non-linear in general, and not the usual operators from linear functionalanalysis; the term ‘linear’ refers to the linearity of the Legendre transformT ∗µ with respect to µ.A consequence of the above definition, is that the expression (2.2) for a10backward linear transfer then becomesT (µ, ν) = supg∈C(Y ){∫Ygdν −∫XT−gdµ}, µ ∈ D1(T ), ν ∈ P(Y ),and (2.3) for a forward linear transfer,T (µ, ν) = supf∈C(X){∫YT+fdν −∫Xfdµ}, µ ∈ P(X), ν ∈ D2(T ).An important observation is that if T is a forward linear transfer, thenT˜ (ν, µ) := T (µ, ν) is a backward linear transfer on P(Y )× P(X), and viceversa. Indeed, if T is a forward linear transfer, thenT˜ ∗ν (f) =∫X−T+(−f)dνso if we define T˜−f := −T+(−f), then we conclude T˜ is a backward lineartransfer on P(Y )× P(X).It therefore suffices to focus either on those T which are backward lineartransfers, or on those T which are forward linear transfers; it will be conve-nient to focus and state results for the backward linear transfers. However,occasionally, we shall see that it is sometimes useful to consider T whichis both a forward, and a backward, linear transfer, hence why we introduceboth notions.We shall often say “T is a backward linear transfer on P(X)×P(Y ) (orP(X)×P(X), P(Y )×P(X), etc.)” for which we mean that T is a backwardlinear transfer with domain P(X)×P(Y ) (or P(X)×P(X), P(Y )×P(X),etc.).Remark 2.3.3. If T happens to be both a forward, and backward, lineartransfer on P(X) × P(X), and is symmetric (i.e. T (µ, ν) = T (ν, µ)), thenT+f = −T−(−f). This in particular, the case for optimal transport (seeSection 2.6).Remark 2.3.4. If T is a backward linear transfer, the definition impliesthe existence of an operator T− satisfying T ∗µ (g) =∫X T−gdµ. Regarding11uniqueness, if there are two such operators T−1 , then∫X T−1 gdµ =∫X T−2 gdµfor all g ∈ C(Y ) and all µ ∈ D1(T ). In particular if {δx ; x ∈ X} ⊂ D1(T )(i.e. the first partial domain of T contains all the Dirac measures), then infact T−1 g(x) = T−2 g(x) for all g and all x. We therefore by abuse of languageshall call in all cases T− the operator associated to T .2.4 Kantorovich operatorsWe now focus our attention on the non-linear operators T−. If T is abackward linear transfer such that D1(T ) contains the Dirac measures, thenits operator T− satisfies (T−g)(x) = T ∗δx(g). Consequently, it inherits certainproperties from the Legendre transform. To this end, we find it convenientto make the following definition. In the following, USC(X) is the collectionof upper semi-continuous functions f : X → R ∪ {−∞}; similarly LSC(X)the collection of g : X → R ∪ {+∞} such that −g ∈ USC(X) (the samenotation holds for Y in place of X).Definition 2.4.1. A backward Kantorovich operator is a map T− :C(Y ) → USC(X) which is proper (i.e. T−g 6≡ −∞ for every g ∈ C(Y ))and verifies the following properties:1. T− is monotone, i.e., if g1 ≤ g2 in C(Y ), then T−g1 ≤ T−g2.2. T− is affine on the constants, i.e., for any c ∈ R and g ∈ C(Y ),T−(g + c) = T−g + c.3. T− is a convex operator, that is for any λ ∈ [0, 1], g1, g2 in C(Y ), wehaveT−(λg1 + (1− λ)g2) ≤ λT−g1 + (1− λ)T−g2.4. T− is lower semi-continuous in the sense that if gn → g in C(Y ) forthe sup norm, then lim infn→∞ T−gn ≥ T−g .For completeness we also define the forward counterpart, although itcan be derived from the backward definition (see the earlier remark on for-12ward linear transfers). In the following, LSC(Y ) denotes the lower semi-continuous functions on Y , i.e. g ∈ LSC(Y ) if and only if −g ∈ USC(Y ).Definition 2.4.2. A forward Kantorovich operator is a map T+ :C(X)→ LSC(Y ) which is proper and verifies1. T+ is monotone, i.e., f1 ≤ f2 in C(X), then T+f1 ≤ T+f2.2. T+ is affine on the constants, i.e., for any c ∈ R and f ∈ C(X),T+(f + c) = T+f + c.3. T+ is a concave operator, that is for any λ ∈ [0, 1], f1, f2 in C(X), wehaveT+(λf1 + (1− λ)f2) ≥ λT+f1 + (1− λ)T+f2.4. T+ is upper semi-continuous, in the sense that if fn → f in C(X) forthe sup norm, then lim supn→∞ T+fn ≤ T+f .Remark 2.4.3. We call these operators ‘Kantorovich’ due to their connec-tion with optimal transport (see Section 2.6). The notion of ‘backward’ isthe interpretation that the operator T− maps a function on Y ‘backward’to a function on X (similarly for the notion of ‘forward’ and T+). The for-ward/backward terminology coincides with Hamilton-Jacobi equations thatare solved forward/backward in time (see Example 2.6.6)We shall see in Section 2.7 that under certain assumptions, backwardlinear transfers turn out to be the dual problems of the weak transportsstudied by Gozlan et al. [36]; for a precise statement, see Theorem 2.7.2.We also mention the recent paper by Alibert, Bouchitte, and Champion[1] which we learned of while preparing [10], in which they study Gozlan’sweak transport costs and their Legendre transform, that we will see cor-respond to Kantorovich operators mapping into the set of bounded uppersemi-continuous functions. We also mention the work of Roos [47] wherethey introduce the notion of viscosity operators as an axiomatic characteri-sation for viscosity solutions to the evolutive Hamilton-Jacobi equation.13We have the following which makes the connection between the backwardlinear transfers and backward Kantorovich operators.Proposition 2.4.4. 1. Suppose T− is a backward Kantorovich operator.Then the function defined byT (µ, ν) :=sup{∫Y gdν −∫X T−gdµ ; g ∈ C(Y )} if (µ, ν) ∈ P(X)× P(Y )+∞ otherwise,is a backward linear transfer provided the expression on the right-handside is finite for at least one (µ, ν) ∈ P(X) × P(Y ). Note that theDirac measures may not be in D1(T ).2. Conversely, suppose T is a backward linear transfer such that the Diracmeasures {δx ; x ∈ X} belong to D1(T ). Then its associated operatorT− in Definition 2.3.1 is a backward Kantorovich operator.In particular T−g(x) is given byT−g(x) = supν∈P(Y ){∫Ygdν − T (δx, ν)},and if g1, g2 are such that T−g1, T−g2 are bounded, then‖T−g2 − T−g2‖∞ ≤ ‖g1 − g2‖∞,where ‖h‖∞ := supx∈X |h(x)|.Proof. 1. Define for µ ∈ D1(T ), Fµ(g) :=∫X T−gdµ, which is necessar-ily finite for all g by choice of µ. Since g 7→ T−g is convex and lowersemi-continuous on C(Y ), it is convex and weakly lower semi-continuous;in particular, this implies Fµ is convex and lower semi-continuous on C(Y ).Therefore, by Fenchel-Moreau, Fµ(g) = supν∈M(Y ){∫Y gdν −F ∗µ(ν)}, whereF ∗µ(ν) = sup{∫Ygdν −∫XT−gdµ ; g ∈ C(Y )}, ν ∈M(Y ).We will be done if we can show that F ∗µ(ν) = T (µ, ν) for all ν ∈ P(Y ). The14equality holds when ν ∈ P(Y ) by definition of T , so suppose ν ∈ M(Y )with ν(Y ) = λ 6= 1. Taking g(x) ≡ n ∈ Z, we have T−(g) = T−(0 + n) =n+ T−(0), and we obtainF ∗µ(ν) ≥ nλ−∫XT−(n)dµ = n(λ− 1)−∫XT−(0)dµ.With n → ±∞, depending on if λ < 1 or λ > 1, we deduce F ∗µ(ν) = +∞.Hence F ∗µ(ν) = T (µ, ν) for all ν ∈M(Y ), and it follows thatFµ(g) = supν∈M(Y ){∫Ygdν − F ∗µ(ν)} = supν∈M(Y ){∫Ygdν − T (µ, ν)} = T ∗µ (g).2. First we note that if {δx ; x ∈ X} ⊂ D1(T ), then in fact T−g ∈USC(X). Indeed, if xn → x in X, extract a subsequence so thatlim supn→∞T−g(xn) = limj→∞T−g(xnj ).From the expressionT−g(x) = T ∗δx(g) = supν∈P(Y ){∫Ygdν − T (δx, ν)}, (2.5)let νj achieve the supremum above when x = xnj (the supremum is achievedby upper semi-continuity of ν 7→ ∫Y gdν − T (δx, ν) on the compact spaceP(Y )). By weak∗ compactness of P(Y ), we may extract a further subse-quence if necessary and assume νj → ν¯ for some ν¯ ∈ P(Y ). It then followsthat by weak∗ lower semi-continuity of Tlim supn→∞T−g(xn) = limj→∞T−g(xnj ) ≤∫Ygdν¯ − T (δx, ν¯)≤ supν∈P(Y ){∫Ygdν − T (δx, ν)} = T−g(x).The remaining properties of monotonicity, convexity, and affine on constantsin Definition 2.3.1 follow immediately from the expression (2.5). For thelower semi-continuity property, suppose gn → g ∈ C(Y ) for the sup norm.15Then from (2.5), we haveT−g(x) = supσ∈P(Y ){∫Ygdσ − T (δx, σ)}≤ supσ∈P(Y ){∫Ygndσ − T (δx, σ)}+ ‖g − gn‖∞= T−gn(x) + ‖g − gn‖∞,henceT−g ≤ lim infn→∞ T−gn(x).Finally, if g1, g2 ∈ C(Y ) are such that T−g1, T−g2 are bounded, then wecan repeat exactly the same estimate as above to haveT−g1(x) ≤ T−g2(x) + ‖g1 − g2‖∞,which, with the assumption that T−g1 and T−g2 are bounded, means wemay subtract and writeT−g1(x)− T−g2(x) ≤ ‖g1 − g2‖∞.Interchanging g1 and g2 yields the 1-Lipschitz estimate for T−.With a further assumption on T , we can ensure that T− maps intoUSCb(X) rather than USC(X) (which in particular, implies T−g is bounded).Proposition 2.4.5. Let T be a backward linear transfer on P(X) × P(Y )such that D1(T ) contains all the Dirac measures. Ifsupx∈Xinfν∈P(Y )T (δx, ν) < +∞,then1. The associated backward Kantorovich operator T− maps C(X) to USCb(X).2. D1(T ) = P(X).16Proof. 1. The only thing which we need to show is that T−g is boundedbelow, since we already know T−g belongs to USC(X) for all g ∈ C(Y ) bythe above proposition. From Proposition 2.4.4, we haveT−g(x) = supν∈P(Y ){∫Ygdν − T (δx, ν)}so that we immediately haveinfx∈XT−g(x) = infx∈Xsupν∈P(Y ){∫Ygdν − T (δx, ν)}≥ inf(g)− supx∈Xinfν∈P(Y )T (δx, ν) > −∞.2. If D1(T ) contains the Dirac measures, then for each x ∈ X, there existsνx ∈ P(Y ), such that T (δx, νx) = infν∈P(Y ) T (δx, ν) by compactness ofY and weak∗ lower semi-continuity of T . Consequently, since T is jointlyconvex then any empirical measure of the form µn :=1n∑ni=1 δxi is in D1(T )withT (µn, νn) ≤ 1nn∑i=1T (δxi , νxi) =1nn∑i=1infν∈P(Y )T (δxi , ν)≤ supx∈Xinfν∈P(Y )T (δx, ν) < +∞,where νn :=1n∑ni=1 νxi .Now let µ ∈ P(X). Since the Dirac measures are the extreme points ofP(X), the measure µ is a weak∗ limit of empirical measures µn := 1n∑ni=1 δxias n → ∞. The measures µn have associated νn such that T (µn, νn) ≤supx∈X infν∈P(Y ) T (δx, ν) < +∞ as above. Then by weak∗ compactness ofP(Y ), we may extract a subsequence so that νnj → ν for some ν ∈ P(Y ).Consequently by weak∗ lower semi-continuity of T , we conclude thatT (µ, ν) ≤ lim infj→∞T (µnj , νnj ) ≤ supx∈Xinfν∈P(Y )T (δx, ν) < +∞.17Proposition 2.4.4 suggests that for an arbitrary map T− from C(Y ) toBorel-measurable functions on X that does not satisfy the conditions for abackward Kantorovich operator, the functionalT (µ, ν) = supg∈C(Y ){∫Ygdν −∫XT−gdµ}may not define a backward linear transfer.Indeed, consider T−g := eg. Then T (µ, ν) := supg∈C(X){∫X gdν −∫X egdµ} can be checked to be proper, and fixing µ ∈ D1(T ), a property ofthe Legendre transform, is that T ∗µ (g+c) = T ∗µ (g)+c for any constant c ∈ Rand all g ∈ C(Y ). Therefore the equality T ∗µ (g) =∫X egdµ cannot hold,since that would imply (taking g ≡ 0 and c = 1), 1 + T ∗µ (0) = T ∗µ (0 + 1) = ewhile at the same time (taking g ≡ 0 and c = 2), it must hold also that2 + T ∗µ (0) = e2, which is impossible. Therefore, we make the followingdefinition.Definition 2.4.6 (Backward linear coupling). T is a backward linearcoupling if there exists a map T− : C(Y )→ Bb(X), such thatT (µ, ν) = supg∈C(Y ){∫Ygdν −∫XT−gdµ}, µ ∈ D1(T ), ν ∈ P(Y ).Remark 2.4.7. Every backward linear transfer is a backward linear cou-pling. On the other hand, note that a backward linear coupling T hasT ∗µ (g) ≤∫X T−gdµ for all µ ∈ D1(T ) and g ∈ C(X).2.5 First examples of linear transfersSo far we have defined the notions of backward (resp., forward) linear trans-fers, but have not provided any examples to suggest the large variety offunctionals which belong to this class. The aim of this section is to highlighta few of the basic ones. As usual, we will present examples on P(X)×P(Y ),with the implicit assumption that they should always be taken identically+∞ outside P(X)× P(Y ) when viewed as a function on M(X)×M(Y ).182.5.1 Convex energiesA class of examples of linear transfers is the convex energies - functionalswhose dependence on the pair (µ, ν) is trivially only through ν.Let I : P(Y ) → R be a bounded below, convex, weak∗-lower semi-continuous function on P(Y ). ConsiderT (µ, ν) := I(ν) for all (µ, ν) ∈ P(X)× P(Y ). (2.6)Then T ∗µ (g) = I∗(g) =∫Y I∗(g)dµ, so T is a backward linear transfer withcorresponding backward Kantorovich operator T−g ≡ I∗(g) (a constantfunction of x). Some choices of I include the following.Example 2.5.1 (The null transfer). This is simply the trivial mappingN (µ, ν) = 0 for all probability measures µ on X and ν on Y , which achievedby taking I ≡ 0 in (2.6). One can then compute I∗(g) = supy∈Y g(y), andit is easy to see that it is both a backward and forward linear transfer withKantorovich operators,T−g ≡ supy∈Yg(y) and T+f ≡ infx∈Xf(x).This is also a particular case of a result from optimal mass transport(which we discuss in Section 2.6) when the cost is identically zero.Example 2.5.2 (Potential energy). If I is the linear functional I(ν) =∫Y V (y) dν(y), where V is a bounded below lower semi-continuous potentialon Y , then for every x ∈ X,T−g ≡ I∗(g) = sup{∫Y(g − V )dν ; ν ∈ P(Y )} = supy∈Y{g(y)− V (y)}.Example 2.5.3 (Relative entropy). Fix any reference measure ν0 ∈ P(Y ),and define I as the relative entropy with respect to ν0, that isIν0(ν) :=∫Ydνdν0log( dνdν0 )dν0 if ν is absolutely continuous with respect to ν0+∞ otherwise,19where dνdν0 is the Lebesgue-Radon-Nikodym derivative of ν with respect toν0, i.e. dν =dνdν0dν0. ThenT−g ≡ I∗(g) = sup{∫Y[gλ− λ log (λ)] dν0 ; λ ≥ 0,∫Yλdν0 = 1}. (2.7)We can actually compute exactly the value of the supremum in (2.7), whichwe state in the following proposition (the following proof is provided in [35],Proposition 2.9 for a more general context).Proposition 2.5.4.T−g ≡ sup{∫Y[gλ− λ log (λ)] dν0 ; λ ≥ 0,∫Yλdν0 = 1} = log∫Yegdν0Proof. Consider for fixed s ∈ R, the function t 7→ st − t log(t) for t ≥ 0.This achieves a maximum supt≥0{st − t log(t)} = es−1. Identify s = g(y)and t = λ(y), we therefore haveλg − λ log λ ≤ eg−1. (2.8)Hence using this inequality (2.8) in (2.7), we have T−g(x) ≤ ∫Y eg−1dν0.From the fact that T−g = T−(g + t) − t for all t ∈ R, it must be the casethatT−g ≤ inft∈R∫Y(eg−1+t − t)dν0.We can now compute the infimum in the inequality: Let F (t) :=∫Y (eg+t−1−t)dν0 for t ∈ R. Since F is convex in t, the minimal value occurs exactlywhere F ′(t0) = 0, i.e. where∫Y eg+t0−1dν0 = 1, which implies that t0 =1− log ∫Y egdν0. This means thatT−g ≤ inft∈RF (t) = F (t0) = log∫Yegdν0.On the other hand, consider the measure ν with density λ = eg∫Y egdν0with20respect to ν0. Then one can check for this λ∫Y[gλ− λ log (λ)] dν0 = log∫Yegdν0giving the reverse inequality T−g ≥ log ∫Y egdν0.Remark 2.5.5. The above shows that the relative entropy functional, de-fined on P(X)× P(X) viaH(µ, ν) =∫Ydνdµ log(dνdµ)dµ if ν is absolutely continuous with respect to µ+∞ otherwiseis not a backward linear transfer, since H∗µ(g) = log∫X egdµ. This motivatesus to introduce a further extension of linear transfers, convex transfers, whichwe will discuss in Section 2.9.Example 2.5.6 (Variance). For the (negative) of the variance,I(ν) := −var(ν) :=∣∣∣∣∫Yy dν(y)∣∣∣∣2 − ∫Y|y|2 dν(y),the associated Kantorovich map is given byT−g ≡ supy∈Y{ĝ + q(y)− |y|2},where q is the quadratic function q(x) = 12 |x|2 and hˆ is the concave envelopeof the function h. The details for this example are given in Example 2.8.4below.2.5.2 Linear transfers induced from positively homogenousoperatorsLet T : C(Y ) → C(X) be a bounded linear positive operator such thatT (1) = 1; we call such an operator a Markov operator. One can associate a21backward linear transfer in the following way:TT (µ, ν) :={0 if ν = T ∗(µ)+∞ otherwise, (2.9)where T ∗ :M(X)→M(Y ) is the adjoint operator. It is immediate to checkthat ν 7→ TT (µ, ν) is convex and weak∗ lower semi-continuous, and(TT )∗µ(g) =∫YgdT ∗(µ) =∫YTgdµ.Therefore T−g := Tg is the corresponding backward Kantorovich operator.Defining pix := T∗(δx), we have that T−f(x) =∫Y f(y)d pix(y) and thatTT (µ, ν) = 0 if and only if ν(B) =∫X pix(B) dµ(x) for any Borel B ⊂ Y .Example 2.5.7 (The prescribed push-forward transfer). Let F be a contin-uous map from X to Y . Given µ ∈ P(X), define the push-forward measureν := F#µ ∈ P(Y ) by ν(B) = µ(F−1(B)) for all Borel sets B ⊂ Y .The operator T−g := g ◦ F is readily seen to be a Markov operator andT ∗(µ) = F#µ; hence (2.9) becomes in this caseTF (µ, ν) :={0 if ν = F#µ,+∞ otherwise (2.10)and the backward Kantorovich operator is given by T−g = g ◦ F . In thecase when X = Y and F (x) = x, TF is both a forward and backward lineartransfer, with T−f = f = T+f , which we call the identity transfer.As a natural extension of the previous example, let A ∈ LSC(X), A 6≡+∞, and consider the Kantorovich operator,T−g(x) := g ◦ F (x)−A(x).22Then the induced backward linear transfer isT (µ, ν) =∫X Adµ if ν = F#µ,∫X Adµ < +∞,+∞ otherwise.We shall see in Section 5.6.1 that this backward linear transfer is relatedto ergodic optimisation in the expanding case [25], where it is of interestto minimise the action µ 7→ ∫X Adµ, for a given potential A, among allF -invariant measures µ (and other related generalisations).Example 2.5.8 (The prescribed Skorokhod transfer). Let X = Y be asmooth compact Riemannian manifold without boundary, and (Bt)t≥0 de-note Brownian motion on X, and S the corresponding class of (possiblyrandomized) stopping times. For a fixed τ ∈ S, defineTτ (µ, ν) :={0 if B0 ∼ µ and Bτ ∼ ν.+∞ otherwise, (2.11)where Z ∼ ρ if Z is a random variable with distribution ρ. It is easy toverify ν 7→ Tτ (µ, ν) is convex and weak∗ lower semi-continuous, and we have(Tτ )∗µ(g) =∫Ygdν = E[g(Bτ )] =∫YE [g(Bτ )|B0 = x] dµ(x)so it is in fact a backward linear transfer with backward Kantorovich oper-ator T−g(x) := E [g(Bτ )|B0 = x].Example 2.5.9 (Prescribed marginals). Let pi be a probability measureon X × Y , and denote pi1 := projX#pi and pi2 := projY #pi as its marginalon X (resp., on Y ). Here we are using the push-forward measure notationof Example 2.5.7, which in this context, means pi1(A) := pi(proj−1X (A)) =pi(A× Y ) (similarly for projY #pi). DefineTpi(µ, ν) ={0 if µ = pi1 and ν = pi2.+∞ otherwise, (2.12)23In this case, pi1 is the only member of D1(Tpi), and(Tpi)∗pi1(g) =∫Ygdpi2 =∫X×Ygdpi =∫X[∫Yg(y)dpix(y)]dpi1(x)where (pix)x is the disintegration of pi with respect to pi1. A similar expressionholds for (Tpi)∗pi2(f). ThereforeT−g(x) :=∫Yg(y)d pix(y) and T+f(y) =∫X f(x)d piy(x).Note in this example that T−g is merely a (bounded above and below)Borel measurable function.2.5.3 Balayage of measuresGiven a closed convex cone A ⊂ C(X) that contains the non-negative con-stant functions and is closed under maxima, define a partial order relationbetween probability measures µ, ν, which we call an A-order, viaµ ≺A ν if and only if∫X ϕdµ ≤∫X ϕdν for all ϕ in A.Define the balayage of measures B on P(X)× P(X) viaB(µ, ν) ={0 if µ ≺A ν+∞ otherwise. (2.13)It is easy to see that ν 7→ B(µ, ν) is convex and weak∗ lower semi-continuous,and thatB∗µ(g) = sup{∫Xgdν ; ν ∈ P(X), µ ≺A ν}.Proposition 2.5.10. We havesup{∫Xgdν ; ν ∈ P(X), µ ≺A ν}=∫X[supδx≺Aσ∫Xg(y)dσ(y)]dµ(x)and hence B is a backward linear transfer.24For the proof, we state the following result that has previously beenestablished in [17] and based on the classical results of Strassen [56]. Wecall a Markov kernel a real-valued function P : P(X) ×X → R, such thatP (·, x) belongs to P(X), and x 7→ P (A, x) is Borel-measurable for any Borelset A ⊂ X.Theorem 2.5.11 ([17], Theorem 2). Let A be a closed convex cone inC(X) which contains the non-negative constant functions and is closed un-der maxima. If µ ≺A ν, then there exists a Markov kernel P , such thatν(A) =∫X P (A, x)dµ(x) for all Borel A ⊂ X, and δx ≺A P (·, x) for allx ∈ X.In addition, the set of extreme points of {(µ, ν) ; µ ≺A ν} is containedin the set of measures (δx, σ), where δx ≺A σ.We also state the following measurable selection theorem, which shall beof use here, and also in later sections.Proposition 2.5.12 ([7], Proposition 7.3.3). Let X be a metrizable space,Y a compact metrizable space, D ⊂ X × Y a closed subset, and f : D →R ∪ {±∞} lower semi-continuous. Let f∗ : projX(D) → R ∪ {±∞}, whereprojX(D) := {x ∈ X ; (x, y) ∈ D for some y ∈ Y }, be defined byf∗(x) = infy∈Dxf(x, y)where Dx := {y ∈ Y ; (x, y) ∈ D}. The f∗ is lower semi-continuous,and there exists a Borel-measurable function ϕ : ProjX(D) → Y such thatGraph(ϕ) ⊂ D and f(x, ϕ(x)) = f∗(x) for all x ∈ ProjX(D).The above results readily leads to the proof of Proposition 2.5.10.Proof. Of Proposition 2.5.10. By Theorem 2.5.11, we conclude that when-ever µ ≺A ν, we have a Markov kernel P with the specified properties, so25that ∫Xg(y)dν(y) =∫X[∫Xg(y)P (dy, x)]dµ(x)≤∫X(supδx≺Aσ∫Xg(y)dσ(y))dµ(x).On the other hand, let σx achieve the supremum for each x ∈ X (the supre-mum is achieved since the set of δx ≺A σ is weak∗ closed in P(X) andtherefore weak∗ compact). Then by the measurable selection Proposition2.5.12, P (A, x) := σx(A) is a Markov kernel, and ν(A) :=∫X P (A, x)dµ(x)satisfies µ ≺A ν.Defining T−g(x) := supδx≺Aσ∫X g(y)dσ(y), we conclude that B is a back-ward linear transfer. We have the following representation for T−:Proposition 2.5.13.T−g(x) = gˆ(x) := inf{h(x) ; h ∈ −A, h ≥ g on X}.Proof. For any admissible h, we have∫X gdσ ≤∫X hdσ, henceT−g(x) = supδx≺Aσ∫Xg(y)dσ(y) ≤ supδx≺Aσ∫Xh(y)dσ(y) ≤ h(x),the last inequality following since h ∈ −A. Hence T−g(x) ≤ gˆ(x).Conversely, let  > 0, and for each x ∈ X, choose h,x ∈ −A, h,x ≥ g,such that h,x(x) ≤ g(x)+ 2 . By continuity, there exists an open neighbour-hood Brx(x) ⊂ X such that h,x(x′) ≤ g(x′) +  for all x′ ∈ Brx(x). Thecollection {Brx(x)}x∈X is an open cover of X, hence by compactness, thereexists a finite subcover {Brxi (xi)}ni=1. Defineh(x) := min{h,x1(x), . . . , h,xn(x)}since A is closed under maxima, it follows that h ∈ −A. Moreover, each26x ∈ X belongs to Brxi (xi) for some i ∈ {1, . . . , n}, henceg(x) ≤ h(x) ≤ g(x) +  for all x ∈ X.Now for any σ such that δx ≺A σ, we have+∫Xgdσ ≥∫Xhdσ ≥ h(x) since h ∈ −A,and since h(x) ≥ gˆ(x), we conclude,+ T−g(x) ≥ +∫Xg(y)dσ(y) ≥ gˆ(x).As  is arbitrary, we obtain the reverse inequality, which concludes the proof.By a similar argument to that given above, one can show that B is alsoa forward linear transfer with a forward Kantorovich operator T+f := fˇ ,wherefˇ(x) = inf{∫Xfdσ; δx ≺A σ} = sup{h(x) ; h ∈ A, h ≤ f on X}.We now discuss particular choices of A.Example 2.5.14 (Convex order). If X is a convex compact space in alocally convex topological vector space, then A can be taken to be the coneof continuous convex functions. In this case, T−g = gˆ is the concave envelopeof g (i.e. the smallest concave function above g), since an infimum of concavefunctions is concave. This is a classical result of Choquet (see e.g. [16]Proposition 26.13).Example 2.5.15 (Subharmonic order). Consider now when X ⊂ Rn and Ais the cone of Lipschitz subharmonic functions on some open set O contain-ing X. In this case, we can identify B in another way using the followingcharacterisation of Skorokhod.Proposition 2.5.16 (Skorokhod, e.g. [30]). Let A be the cone of Lips-27chitz subharmonic functions on a domain X in Rn. Then, the following areequivalent for two probability measures µ and ν on X.1. µ ≺A ν.2. There exists a stopping time τ ∈ S with E[τ ] < ∞ such that B0 ∼ µand Bτ ∼ ν, where S denote the collection of (random) Brownianstopping times with finite expectation.Therefore,B(µ, ν) =0 if B0 ∼ µ and Bτ ∼ ν for some τ ∈ S,+∞ otherwise.This identification of B provides various ways to write T− in this case:1. T−g = gˆ is the smallest Lipschitz superharmonic function above g.2. T−g = supτ∈SE [g(Bτ )|B0 = x].3. T−g = Jg, where Jg(x) is a viscosity solution for the heat variationalinequality max {g(x)− J(x),∆J(x)} = 0.Note also that B(µ, ν) = infτ∈STτ (µ, ν) where Tτ is the “prescribed Skorokhodtransfer” of Example 2.5.8.2.6 Linear transfers and optimal masstransportationIn the Monge-Kantorovich theory of optimal transport (see e.g. [59, 60], or[48]), given a lower semi-continuous cost function c : X × Y → R ∪ {+∞}one associates a map Tc on P(X)×P(Y ) to be the optimal total transportcost between µ ∈ P(X) and ν ∈ P(Y ), viaTc(µ, ν) := infpi∈K(µ,ν)∫X×Yc(x, y) dpi(x, y), (2.14)28where K(µ, ν) is the set of probability measures pi on X×Y whose marginalprojX]pi on X is µ, and marginal projY ]pi on Y is ν, (recall projX is theprojection (x, y) 7→ x onto X; similarly for Y ). Also recall the push-forwardof a measure by a map φ is the measure (φ#)µ(A) := µ(φ−1(A)).The measures pi ∈ K(µ, ν) are interpreted as transport plans, with dpi(x, y)representing the amount of mass which is sent from x to y, and costingc(x, y)dpi(x, y) to do so.The optimal transport problem (2.14) is an infinite dimensional linearprogramming problem; as in the classical finite dimensional theory, it has adual formulation first studied by Kantorovich (see e.g. [59], Theorem 1.3,and references therein).Theorem 2.6.1 (Kantorovich duality [59]). Let c : X × Y → R∪ {+∞} bea lower semi-continuous function. Theninfpi∈K(µ,ν)∫X×Yc(x, y) dpi(x, y) = sup{∫Ygdν −∫XT−gdµ ; g ∈ C(Y )}= sup{∫YT+fdν −∫Xfdµ ; f ∈ C(X)}whereT−g(x) := supy∈Y{g(y)− c(x, y)} and T+f(y) := infx∈X{c(x, y) + f(x)}.Remark 2.6.2. If X and Y are not compact, then c should also be assumedto be bounded below; this additional hypothesis is not necessary whenX andY are compact as c is automatically bounded from below by the assumptionof lower semi-continuity.Proof. Note first that ν 7→ Tc(µ, ν) is convex and weak∗ lower semi-continuous.Indeed, for ν1, ν2 ∈ P(Y ), and fixed µ ∈ P(X), the infimum is achievedfor Tc(µ, νi) at some pii ∈ K(µ, νi), i = 1, 2. It immediately follows thatpiλ := (1− λ)pi1 + λpi2 is admissible for Tc(µ, (1− λ)ν1 + λν2). In addition,if νn → ν, then for any pin ∈ K(µ, νn), one may extract a subsequence con-verging to pi ∈ P(X×Y ) by weak∗ compactness of P(X×Y ), and moreover,29pi ∈ K(µ, ν). The lower semi-continuity then follows. Finally,(Tc)∗µ(g) = supν∈P(Y ){∫Ygdν − infpi∈K(µ,ν)∫X×Yc(x, y) dpi(x, y)}= supν∈P(Y )suppi∈K(µ,ν){∫X×Y(g(y)− c(x, y))dpi(x, y)}= suppi∈K(µ,·){∫X×Y(g(y)− c(x, y))dpi(x, y)}.It is clear that the supremum is achieved by a pi whose disintegration withrespect to µ is given by dpi(x, y) = dµ(x)δyx(y) where yx is a value of yachieving supy∈Y {g(y)− c(x, y)}. A similar argument holds for showing Tcis a forward linear transfer.Remark 2.6.3. In fact one can refine the duality further and show thatinfpi∈K(µ,ν)∫X×Yc(x, y) dpi(x, y) = sup{∫Yφ1dν −∫Xφ0dµ}where the supremum is over all pairs (φ0, φ1) such that φ1 = T−φ0 and φ0 =T+φ1. This will be relevant for Bernard-Buffoni’s work [5, 6] discussed inChapter 4 on optimal transport and Aubry-Mather and weak KAM theory.In the following examples, we highlight the backward/forward Kan-torovich operators for a few special cases for the cost function.Example 2.6.4 (The trivial Kantorovich transfer). Let c1 ∈ USCb(X),c2 ∈ LSC(Y ), and define c(x, y) := c2(y) − c1(x). Then c is lower semi-continuous, andTc(µ, ν) =∫Yc2 dν −∫Xc1 dµ.The Kantorovich operators are T+f = c2 +infx∈X{f(x)−c1(x)} and T−g =c1 + supy∈Y {g(y)− c2(y)}.Example 2.6.5 (The Kantorovich-Rubinstein transport). For the cost c(x, y) =dX(x, y) (recall dX is the metric on X), the Kantorovich operators are given30byT+f(y) = infx∈X{f(x)+dX(x, y)} and T−g(x) = supy∈X{g(y)−dX(x, y); y ∈ X}.A computation shows that T− ◦ T+ = T+ and T+ ◦ T− = T−. Hence bytaking f = T−h for h ∈ C(X), we haveTc(µ, ν) = supf∈C(X){∫XT+fdν −∫Xfdµ} ≥ suph∈C(X){∫XT−h d(ν − µ)}while at the same time, from g ≤ T−g we haveTc(µ, ν) = supg∈C(X){∫Xgdν −∫XT−gdµ} ≤ supg∈C(X){∫XT−g d(ν − µ)}.Hence, Tc(µ, ν) = supg∈C(X){∫X T−g d(ν − µ)}. It is readily checked thatT−g is 1-Lipschitz for any g ∈ C(X), and moreover, every g which is λ-Lipschitz, λ ≤ 1, satisfies T−g = g. Therefore,T (µ, ν) = sup{∫Xg d(ν − µ); g ∈ C(X), ‖g‖Lip ≤ 1}where ‖g‖Lip := supx 6=y |g(y)−g(x)|dX(x,y) . (The above discussion holds also for anyother lower semi-continuous metric d, not necessarily the one defining thetopology.)Example 2.6.6 (Lagrangian cost [5]). This example links the Kantorovichbackward and forward operators with the respective backward and forwardHopf-Lax operators that solve first order Hamilton-Jacobi equations, whichwe will see is intimately connected to weak KAM theory (see Chapter 4).On a given smooth compact Riemannian manifold M without boundary(take e.g. M = Tn the flat torus), with tangent bundle TM , let L : TM →R ∪ {+∞} be a given Tonelli Lagrangian (see Chapter 4 for a definition),and consider the costcL(y, x) := inf{∫ 10L(γ(t), γ˙(t)) dt; γ ∈ C1([0, 1],M); γ(0) = y, γ(1) = x}.31For µ and ν two probability measures on M , consider the optimal transportwith cost cL.Then, T+f(x) = Vf (1, x), where Vf (t, x) is the “value functional”Vf (t, x) := inf{f(γ(0)) +∫ t0L(γ(s), γ˙(s)) ds; γ ∈ C1([0, 1],M), γ(t) = x}.Vf is in fact a viscosity solution for the Hamilton-Jacobi equation{∂tV +H(x,∇xV ) = 0 on (0,+∞)×M,V (0, x) = f(x).(see e.g. [20], Theorem 7.2.8) where H is the Hamiltonian (see Chapter4). Similarly, the backward Kantorovich operator is given by T−g(y) =Wg(0, y), where Wg(t, y) is the value functionalWg(t, y) := sup{g(γ(1))−∫ 1tL(γ(s), γ˙(s)) ds; γ ∈ C1([0, 1),M); γ(t) = y},which is a viscosity solution for the backward Hamilton-Jacobi equation{∂tW +H(x,∇xW ) = 0 on [0, 1)×M,W (1, y) = g(y).Example 2.6.7 (The Brenier-Wasserstein distance [12]). We mention thisimportant example even though it is not in a compact setting. If c(x, y) =〈x, y〉 on Rd×Rd, and µ, ν are two probability measures of compact supporton Rd, thenT (µ, ν) = inf {∫Rd×Rd〈x, y〉 dpi;pi ∈ K(µ, ν)}.Here, the Kantorovich operators areT+f(x) = −f∗(−x) and T−g(y) = (−g)∗(−y),where f∗ is the convex Legendre transform of f .322.7 Representation and envelopes of lineartransfers2.7.1 Linear transfers and optimal weak transportsA natural generalisation of optimal transport is optimal weak transport, in-troduced by Gozlan et. al. [36]. Weak transport is concerned with costfunctions c(x, σ) between points x and distributions σ ∈ P(Y ), and thus isa generalisation of the cost-minimizing optimal transport of the last sectionwhich concerns cost functions between points x and y.Definition 2.7.1. Let c : X × P(Y ) → R ∪ {+∞} be a bounded below,lower semi-continuous function with σ 7→ c(x, σ) convex. The optimalweak transport problem with cost c from µ ∈ P(X) to ν ∈ P(Y ) isV (µ, ν) := infpi{∫Xc(x, pix) dµ(x);pi ∈ K(µ, ν)} (2.15)where (pix)x is the disintegration of pi with respect to µ.This problem has been the study of several recent papers [1], [3], and[36], where existence, duality, and certain properties of optimisers of (2.15)have been established. In the next theorem we show that weak transportsare backward linear transfers, and, conversely, appropriate backward lineartransfers can be represented as weak transports. The implication of this isthat, roughly speaking, backward linear transfers are the dual formulationof weak transports.Theorem 2.7.2. 1. Any optimal weak transport cost c(x, σ) satisfyingthe properties in Definition 2.7.1 defines a backward linear transferviaT (µ, ν) :=inf{∫X c(x, pix) dµ(x);pi ∈ K(µ, ν)}, if (µ, ν) ∈ P(X)× P(Y ),+∞ otherwise,and T−g(x) := sup{∫Y gdσ − c(x, σ) ; σ ∈ P(Y )}.332. Conversely, if T is a backward linear transfer such that {δx;x ∈ X} ⊂D1(T ), then T is an optimal weak transport with cost function c(x, σ) :=T (δx, σ).In order to prove that a weak transport V (µ, ν) is a backward lineartransfer, we wish to show convexity and weak∗ lower semi-continuity for ν 7→V (µ, ν). It turns out that the weak∗ lower semi-continuity is more delicateto establish than standard optimal transport due to the disintegration pix ofpi with respect to its first marginal; two recent papers, [1] (see also [3]) haveclosed this gap.Theorem 2.7.3 ([1], Theorem 2.9, Lemma 3.5). Let c satisfy the conditionsgiven in Definition 2.7.1. Then the map pi 7→ I(pi) := ∫X c(x, pix)dµ(x) isweak∗ lower semi-continuous, where µ := ProjX#pi is the first marginal ofpi. Consequently, the problem (2.15) admits a minimiser.With these properties, we give the rest of the details for the proof ofTheorem 2.7.2.Proof. (Of Theorem 2.7.2)1. We show that ν 7→ T (µ, ν) is convex. This is essentially a consequenceof the assumption that σ 7→ c(x, σ) is convex. Indeed, fix ν1, ν2 ∈P(Y ), and find for a fixed  > 0, pi1 ∈ K(µ, ν1) and pi2 ∈ K(µ, ν2) suchthat ∫Xc(x, piix) dµ(x) ≤ Tc(µ, νi) +  for i = 1, 2.Consider pi ∈ P(X × Y ) which is defined viadpi(x, y) := (λdpi1x(y) + (1− λ)dpi2x(y))dµ(x).With νλ := λν1 + (1 − λ)ν2, we have pi ∈ K(µ, νλ), and therefore, byconvexity of c in the second variable, we haveTc(µ, νλ) ≤∫Xc(x, pix) dµ(x) ≤∫Xλc(x, piix) dµ(x) +∫X(1− λ)c(x, piix) dµ(x)≤ λTc(µ, ν1) + (1− λ)Tc(µ, ν2) + ,34which implies convexity of ν 7→ Tc(µ, ν).We now show that ν 7→ T (µ, ν) is weak∗ lower semi-continuous. Thisis almost immediate from Theorem 2.7.3. Indeed, let νn → ν andselect a subsequence so that limj→∞ T (µ, νnj ) = lim infn→∞ T (µ, νn).For notational ease, relabel the subsequence to νn. For each n, letpin ∈ K(µ, νn) be optimal for the weak transport T (µ, νn), which weknow exists by Theorem 2.7.3. By compactness of P(X × Y ), there isa further subsequence (which we again relabel to n) so that pin → pifor some pi ∈ K(µ, ν). This pi is admissible for T (µ, ν) soT (µ, ν) ≤∫Xc(x, pix) dµ(x)while at the same time by Theorem 2.7.3, we have∫Xc(x, pix) dµ(x) ≤ lim infn→∞∫Xc(x, pinx) dµ(x) = lim infn→∞ T (µ, νn)which concludes the proof of lower semi-continuity. Finally, we com-pute the Legendre transform,T ∗µ (g) = supν∈P(Y ){∫Yg dν − T (µ, ν)}= supν∈P(Y )suppi∈K(µ,ν){∫Yg(y) dν(y)−∫Xc(x, pix) dµ(x)}= suppi∈K(µ,·){∫X[∫Yg(y)dpix(y)− c(x, pix)]dµ(x)} (2.16)≤∫Xsupσ∈P(Y ){∫Ygdσ − c(x, σ)}dµ(x) (2.17)=∫XT−gdµ.On the other hand, by compactness of P(Y ) and lower semi-continuityof c, for each x the supremum in (2.17) is achieved by some σx in ameasurable way (see the measurable selection proposition 2.5.12) so35thatT−g(x) =∫Yg(y)dσx(y)− c(x, σx) for every x ∈ X.Define pi ∈ P(X × Y ) via dpi(x, y) := dσx(y)dµ(x). Denoting ν :=ProjY #pi, we have pi ∈ K(µ, ν). Hence, this pi is admissible in thesupremum (2.16), so thatT ∗µ (g) ≥∫Y[∫Xg(y)dσx(y) dµ(x)−∫Xc(x, σx)]dµ(x)=∫XT−g(x)dµ(x),hence T ∗µ (g) =∫X T−g(x)dµ(x).2. Let T be a backward linear transfer with backward Kantorovich oper-ator T−. Define c(x, σ) := T (δx, σ). Then c is bounded below, proper,lower semi-continuous, and σ 7→ c(x, σ) is convex, so we may definethe weak transportT˜ (µ, ν) := infpi∈K(µ,ν)∫Xc(x, pix) dµ(x)for µ ∈ P(X) and ν ∈ P(Y ) (and +∞ otherwise on M(X)×M(Y )).In an identical way to the proof of item 1, we have for any µ,T˜ ∗µ (g) =∫XT˜−gdµwhere T˜−g(x) := supσ∈P(Y ){∫Y gdσ − c(x, σ)}. On the other hand,since c(x, σ) = T (δx, σ) and {δx ; x ∈ X} ⊂ D1(T ), we also have thatT˜−g(x) = T ∗δx(g) = T−g(x) for all x ∈ X. Hence T˜ ∗µ (g) =∫X T−gdµ =T ∗µ (g) for all µ ∈ P(X) and consequently, T (µ, ν) = T˜ (µ, ν).362.7.2 Linear transfer envelopesSuppose T is merely a proper, convex, bounded below, weak∗ lower semi-continuous functional, but is not a backward linear transfer, or even a back-ward linear coupling. Is there a “canonical” backward linear transfer associ-ated to T ? We therefore consider in the following linear transfer “envelopes”,in an analogy to the convex/concave envelopes of functions.Proposition 2.7.4. Let T : P(X) × P(Y ) → R ∪ {+∞} be a proper,convex, bounded below, weak∗ lower semi-continuous functional such that{δx ; x ∈ X} ⊂ D1(T ). Denote T to be the weak transport defined by (2.15)associated to the weak transport cost c(x, σ) := T (δx, σ), and letT˜ (µ, ν) :=∫Xc(x, ν)dµ(x).Then, T ≤ T ≤ T˜ , and T is the only weak transport between T and T˜ .In particular, the following holds: For any backward linear transfer Swith {δx ; x ∈ X} ⊂ D1(S),1. If T ≤ S, then T ≤ S.2. If S ≤ T˜ , then S ≤ T .Proof. Note that by the first part of Theorem 2.7.2, T is a backward lineartransfer with backward Kantorovich operatorT−g(x) = supσ∈P(Y ){∫Yg dσ − c(x, σ)}.Moreover, as T is a weak transport, an admissible pi ∈ K(µ, ν) is the productmeasure µ⊗ ν. HenceT (µ, ν) ≤∫Xc(x, ν)dµ(x) = T˜ (µ, ν).To show that T ≤ T , note that since T is jointly convex and lower semi-37continuous, then for each g ∈ C(Y ), the functionalµ→ (Tµ)∗(g) = supσ∈P(Y ){∫Ygdσ − T (µ, σ)}is upper semi-continuous and concave. It follows from Jensen’s inequalitythat(Tµ)∗(g) ≥∫X(Tδx)∗(g)dµ(x) =∫XT−g(x)dµ(x),henceT (µ, ν) = (Tµ)∗∗(ν) ≤ supg∈C(Y ){∫Ygdν −∫XT−gdµ}= T (µ, ν).To see that T is the smallest backward linear transfer greater than T ,note that if S is a backward linear transfer and T ≤ S, then T ≤ S, andS = S by the previous Theorem 2.7.2.To see T is the greatest backward linear transfer smaller than T˜ , sup-pose S is a backward linear transfer with S ≤ T˜ and S− is its backwardKantorovich operator. Note that T (δx, ν) = T˜ (δx, ν). ThereforeS−g(x) = S∗δx(g) ≥ T˜ ∗δx(g) = T∗δx(g) = T−g(x),and therefore S ≤ T .Remark 2.7.5. Suppose T is any convex, bounded below, and weak∗ lowersemi-continuous functional on P(X)×P(Y ) that is finite on the set of Diracmeasures {(δx, δy) ; x ∈ X, y ∈ Y }. One can then define a cost functionc(x, y) = T (δx, δy), and the associated optimal transport Tc(µ, ν). To com-pare T with Tc, note thatT ∗δx(g) = sup{∫Ygdν − T (δx, ν); ν ∈ P(Y )}≥ sup{g(y)− c(x, y); y ∈ Y } = T−c g(x)38and soT (µ, ν) ≤ T (µ, ν) ≤ Tc(µ, ν).In many cases, it is not possible to define a proper cost c(x, y) = T (δx, δy),i.e. c is identically +∞. This is the case for many stochastic transportproblems where transport via Brownian motion makes it impossible for aDirac measure to be transported to another Dirac measure; see in particularthe stochastic mass transport of Example 2.8.7.In the next proposition, we describe dually a “Kantorovich operatorenvelope” for operators T : C(Y ) → USC(X). It will be necessary first(and later in this thesis) to state a classical minimax theorem that willallow us to interchange sup and inf in appropriate cases.Theorem 2.7.6 (Sion’s minimax theorem [52], Theorem 3.4). Let V be acompact convex subset of a topological vector space, and W a convex set ina (possibly different) topological vector space. Let f : V ×W → R be suchthat1. For all w ∈ W , v 7→ f(v, w) is: (i) lower semi-continuous on V and(ii) quasiconvex on V , i.e. {v ∈ V ; f(v, w) ≤ λ} is convex or emptyfor all λ ∈ R,2. For all v ∈ V , w 7→ f(v, w) is: (i) upper semi-continuous on W and(ii) quasiconcave on W , i.e. {w ∈ W ; f(v, w) ≥ λ} is convex orempty for all λ ∈ R,Then infv∈V supw∈W f(v, w) = supw∈W infv∈V f(v, w).Proposition 2.7.7. Let T : C(Y ) → USCb(X) be any map such that forevery x ∈ X,supx∈Xinfν∈P(Y )supg∈C(Y ){∫Yg dν − Tg(x)} < +∞. (2.18)Then the operator defined byT−g(x) := supσ∈P(Y )infh∈C(Y ){∫Y(g − h) dσ + Th(x)}39is the backward Kantorovich operator of the weak transport T¯ associated toT of Proposition 2.7.4 and maps C(Y ) to USCb(X). It satisfies T− ≤ T ,with the property that for any other backward Kantorovich operator S− :C(Y )→ USCb(X) such that S− ≤ T , then S− ≤ T−. Consequently if T isitself a backward Kantorovich operator, then T−= T .Proof. Consider the backward linear coupling on M(X)×M(Y ) given byT (µ, ν) ={supg∈C(Y ){ ∫Y g dν −∫X Tg dµ}if µ, ν ∈ P(X)× P(Y ),+∞ otherwise.Note that T is a bounded below, convex, lower semi-continuous functionaland the assumption (2.18) means that {δx ; x ∈ X} ⊂ D1(T ). Hence Propo-sition 2.7.4 applies to yield a weak transport T with a corresponding back-ward Kantorovich operator defined asT−g(x) = supσ∈P(Y ){∫Ygdσ − T (δx, σ)}.We can then provide an upper estimate for T−g(x) by substituting theexpression for T (δx, σ):T−g(x) = supσ∈P(Y ){∫Ygdσ − T (δx, σ)}= supσ∈P(Y )infh∈C(Y ){∫Ygdσ −∫Yhdσ + Th(x)}≤ infh∈C(Y )supσ∈P(Y ){∫gdσ −∫hdσ + Th(x)}= infh∈C(Y ){sup(g − h) + Th(x)}≤ Tg(x).40If S is a backward Kantorovich operator such that S ≤ T , thenT−g(x) = supσ∈P(Y )infh∈C(Y ){∫Ygdσ −∫Yhdσ + Th(x)}≥ supσ∈P(Y )infh∈C(Y ){∫gdσ −∫hdσ + Sh(x)}= infh∈C(Y )supσ∈P(Y ){∫gdσ −∫hdσ + Sh(x)}= infh{sup(g − h) + Sh(x)}= infh{S[sup(g − h) + h](x)}≥ Sg(x).where the last two steps used the fact that S is a Kantorovich operator.2.7.3 Recessions of linear transfersWe recall in classical convex analysis, a direction of recession for a convexset C ⊂ X is a direction d such that x+λd ∈ C for all x ∈ C and all λ ≥ 0.The recession cone is the collection of all such d. The recession cone for aconvex function f is the intersection of all recession cones for its sublevelsets. One can introduce the recession function of f asrf (d) := limλ→+∞f(x+ λd)− f(x)λwhich characterises the recession cone for f as the collection d where rf (d) ≤0.In an analogy, we have the following recession operator for linear trans-fers, which was introduced in [1].Proposition 2.7.8. Let T be a backward linear transfer such that {δx;x ∈X} ⊂ D1(T ), with backward Kantorovich operator T−. Then, the functionalTfinite(µ, ν) ={0 if T (µ, ν) < +∞+∞ otherwise,41is a backward linear transfer with Kantorovich operatorT−r g(x) := limλ→+∞T−(λg)(x)λ= sup{∫Ygdν ; ν ∈ P(Y ), T (δx, ν) < +∞}. (2.19)Proof. Without loss of generality, we may assume T ≥ 0 (otherwise considerT −C where C ∈ R is a lower bound for T ). It is immediate that Tfinite is aproper, bounded below, convex, and weak∗ lower semi-continuous functionwith D1(Tfinite) = D1(T ). We have for µ ∈ D1(Tfinite),(Tfinite)∗µ(g) = sup{∫Ygdν ; ν ∈ P(Y ), T (µ, ν) < +∞}. (2.20)At the same time, we know by Proposition 2.4.4 that T− is given byT−g(x) = supσ∈P(Y ){∫Yg dσ − T (δx, σ)},so for λ > 0, we haveT−(λg)(x)λ= supσ∈P(Y ){∫Yg dσ − 1λT (δx, σ)}.We then have lim infλ→+∞T−(λg)(x)λ ≥ sup{∫Y gdσ ; T (δx, σ) < +∞}. Onthe other hand, since T is non-negative, we haveT−(λg)(x)λ≤ sup{∫Yg dσ ; σ ∈ P(Y ), T (δx, σ) < +∞}and we conclude thatlimλ→+∞T−(λg)(x)λ= sup{∫Yg dσ ; σ ∈ P(Y ), T (δx, σ) < +∞}. (2.21)Exactly the same argument also shows that for any µ ∈ D1(T ),limλ→+∞∫XT−(λg)λdµ = sup{∫Ygdν ; ν ∈ P(Y ), T (µ, ν) < +∞}. (2.22)42To conclude that Tfinite is a backward linear transfer, it remains then tojustify limλ→+∞∫XT−(λg)λ dµ =∫X limλ→+∞T−(λg)λ dµ. This is simply bymonotone convergence. Indeed, as T is non-negative, then T−(λg)(x)λ is mono-tone increasing in λ; if λ2 ≥ λ1,T−(λ2g)(x)λ2= supν∈P(Y ){∫Ygdν − 1λ2T (δx, ν)}≥ supν∈P(Y ){∫Ygdν − 1λ1T (δx, ν)} = T−(λ1g)(x)λ1.We therefore conclude by (2.20), (2.21), and (2.22), that(Tfinite)∗µ(g) =∫XT−r g dµwhere T−r g(x) is given by (2.19).Remark 2.7.9. Note that the above impliesT (µ, ν) < +∞ if and only if ∫X T−r gdµ ≥ ∫Y gdν for every g ∈ C(Y ).The latter condition can be seen as a generalized order condition betweenµ and ν. For example, if T is the balayage transfer of Section 2.5.3 onP(X) × P(X) with A = C the cone of convex functions, then T−r g = gˆ,the concave envelope of g. In this case, if T (µ, ν) = 0, then µ C ν, i.e.∫X ϕdµ ≤∫X ϕdν for all convex functions ϕ. Hence for all g ∈ C(X),∫Xgdν ≤∫Xgˆdν ≤∫Xgˆdµ =∫XT−r gdµ.On the other hand, if∫X gˆdµ =∫X T−r gdµ ≥∫X gdν for every g ∈ C(X),the inequality holds for all concave η, and in this case ηˆ = η. This impliesµ C ν and consequently T (µ, ν) = 0.432.8 Linear transfers which are not masstransportsWe now give examples of linear transfers, which do not fit in the frameworkof Monge-Kantorovich theory of Section 2.6.2.8.1 Linear transfers associated to weak mass transportsExample 2.8.1 (Marton transports [41, 42] are backward linear transfers).Consider the weak costc(x, σ) := γ(∫Yd(x, y)dσ(y))where γ is a convex function on R+ and d : X × Y → R is a lower semi-continuous function. The associated weak transport is a backward lineartransfer with Kantorovich operatorT−g(x) = supσ∈P(Y ){∫Ygdσ − γ(∫Yd(x, y) dσ(y))}.Proposition 2.8.2. Let T be a backward linear transfer on P(X)×P(Y ),where Y is convex and compact. Suppose for some lower semi-continuousfunctional c : X × Y → R, we haveT (δx, σ) = c(x,∫Yy dσ(y)) for all x ∈ X and σ ∈ P(Y ),where∫Y y dσ(y) denotes the barycentre of σ. Then, for every g ∈ C(Y ),T−g(x) = supy∈Y{gˆ(y)− c(x, y)},where gˆ is the concave envelope of g (see Example 2.5.14).Proof. Note that z is the barycenter of a probability measure σ if and only44if δz ≺C σ where C is the cone of convex functions. Write nowT−g(x) = supσ∈P(Y ){∫Yg dσ − c(x,∫Yy dσ(y))}= supz∈Y{sup{∫g dσ;σ ∈ P(Y ), δz ≺C σ} − c(x, z)}= supz∈Y{gˆ(z)− c(x, z)}where the last equality we have used Proposition 2.5.13.Example 2.8.3 (A barycentric cost function (Gozlan et al. [36])). Considerthe weak transportT (µ, ν) = inf{∫X‖x−∫Yydpix(y)‖ dµ(x);pi ∈ K(µ, ν)}where ‖ · ‖ is the norm on X induced from dX . This is a backward lineartransfer, with Kantorovich operator given byT−gf(x) = sup{gˆ(y)− ‖y − x‖; y ∈ Y },where gˆ is the concave envelope of g.Example 2.8.4 (The variance functional). Consider the variance functionalT (µ, ν) := |∫Yy dν|2 −∫Y|y|2 dν(y).Denoting q(y) = |y|2 as the quadratic function, we haveT−f(x) = sup{∫Yf dσ − |∫Yy dσ|2 +∫Y|y|2 dσ(y);σ ∈ P(Y )}= sup{∫Y(f + q) dσ − |∫Yy dσ|2;σ ∈ P(Y )}=S−(f + q)(x),where S− is the Kantorovich operator associated to the transfer S(µ, ν) :=| ∫Y y dσ|2, which only depends on the barycenter and therefore S−g =45sup{gˆ(z)− |z|2; z ∈ Y }. It follows thatT−f = sup{f̂ + q(z)− |z|2; z ∈ Y }.Cost minimizing mass transport with additional constraints give exam-ples of one-directional (backward) linear transfers.Example 2.8.5 (Martingale transport). Let c : X×X → R be a continuouscost function and C the cone of convex functions. Martingale transport (seee.g. [29], [37]) is defined byT (µ, ν) =inf{∫X×X c(x, y)dpi(x, y) ; pi ∈M(µ, ν)} if µ ≺C ν+∞ otherwise,where M(µ, ν) consists of all transport plans pi ∈ K(µ, ν) such that theirdisintegration pix with respect to µ, dpi(x, y) = dµ(x)dpix(y), has barycentrex.Note that pix has barycentre x, if and only if, δx ≺C pix. Therefore,defining the weak costc˜(x, σ) ={ ∫X c(x, y) dσ(y) if δx ≺C σ,+∞ if not,martingale transport can be written asT (µ, ν) =infpi∈K(µ,ν){∫X×X c˜(x, pix) d µ} if µ ≺C ν,+∞ otherwise.Since it is a weak transport, we know from Section 2.7 that it is a backward46linear transfer withT−g(x) = supσ∈P(X){∫Xgdσ − c˜(x, σ)}= sup{∫Y(g(y)− c(x, y))dσ(y) ; σ ∈ P(Y ), δx C σ}= gˆc,x(x),where gˆc,x is the concave envelope of the function y 7→ g(y) − c(x, y). Thelast equality follows from Proposition 2.5.13.Example 2.8.6 (Schro¨dinger bridge (Gentil-Leonard-Ripani [27])). Thisexample is in Rd, but we include it even though we have not formally definedlinear transfers on non-compact spaces. Fix some reference non-negativemeasure R on path space Ω = C([0, 1],Rd). Let X = (Xt)t be a randomprocess on M whose law is R, i.e. R = (ΦX)#(P), where (ΦX(ω))(t) :=Xt(ω). Denote by R01 the joint law of the initial position X0 and the finalposition X1. For example (see [27]), assume R is the reversible Kolmogorovcontinuous Markov process associated with the generator 12(∆ − ∇V · ∇)and the initial measure m = e−V (x)dx for some function V .For probability measures µ and ν on M , defineTR01(µ, ν) := inf{∫RdH(rx1 , pix)dµ(x) ; pi ∈ K(µ, ν), dpi(x, y) = dµ(x)dpix(y)}where dR01(x, y) = dm(x)drx1 (y) is the disintegration of R01 with respect toits initial measure m. We see that TR01 is a weak transport correspondingto the weak cost c(x, p) = H(rx1 , p) and hence, by an appropriate extensionof Theorem 2.7.2 to non-compact spaces, it is a backward linear transfer.Its backward Kantorovich operator is given byT−f(x) = logERxef(X1) = logS1(ef )(x),where (St) is the semi-group associated to R.The transfer (2.8.6) is associated to the maximum entropy formulationof the Schro¨dinger bridge problem in the following way: Define the entropic47transportation cost between µ and ν via the formulaSR(µ, ν) = inf{∫Rd×Rdlog(dpidR01) dpi;pi ∈ K(µ, ν)}.Then, under appropriate conditions on V (e.g., if V is uniformly convex),thenTR01(µ, ν) = SR(µ, ν)−∫Mlog(dµdm) dµ.Note that when V = 0, the process is Brownian motion with Lebesgue mea-sure as its initial reversing measure, while when V (x) = |x|22 , R is the pathmeasure associated with the Ornstein-Uhlenbeck process with the Gaussianas its initial reversing measure.2.8.2 One-sided transfers associated to stochastic masstransportLet M = Tn be the flat torus, and consider a Lagrangian on phase space L :TM → [0,∞). Let (Ω,F , (Ft)t,P) be a complete filtered probability spaceand define A[0,1] to be the set of continuous semi-martingales X : Ω×[0, 1]→M such that there exists a Borel measurable drift βX : [0, 1]×C([0, 1])→ Rdfor which1. ω 7→ βX(t, ω) is B(C([0, t]))+-measurable for all t ∈ [0, 1], whereB(C([0, t])) is the Borel σ-algebra of C[0, t].2. WX(t) := X(t) − X(0) −∫ t0 βX(s,X)ds is a σ(X(t) ; 0 ≤ t ≤ 1) M -valued Brownian motion.Example 2.8.7 (Stochastic mass transport between two probability mea-sures). Consider the functional T : P(M) × P(M) → R ∪ {+∞} definedbyT (µ0, µ1) := inf{E∫ 10L(X(s), βX(s,X))ds ; X(0) ∼ µ0, X(1) ∼ µ1, X ∈ A[0,1]},where we use the notation Z ∼ σ to denote that the random variable Z hasdistribution σ.48This stochastic transport does not fit in the standard optimal mass trans-port theory since it does not originate in the optimization according to acost between two deterministic states. However, it still enjoys a dual formu-lation (first proven by Mikami-Thieullin [45] for the space Rn) that permitsit to be realised as a backward linear transfer. An adaptation of the proofsof Mikami-Thieullin to M = Tn (see [45] Lemma 3.1, 3.2, and 3.3, andTheorem 2.2), yield the following.Proposition 2.8.8. Under suitable conditions on L (for example, if L(x, β) =12 |β|2; see in [45] the assumptions (A1), (A2), (A3)(i - v), and (A4)),ν 7→ T (µ, ν) is convex and weak∗ lower semi-continuous, andT (µ, ν) = supg∈C(M){∫Mg(x)dν(x)−∫Mug(0, x)dµ(x)}where ug : [0, 1]×M → R is a viscosity solution of∂u∂t(t, x) +12∆xu(t, x) +H(x,∇xu(t, x)) = 0, (t, x) ∈ [0, 1)×Mwith u(1, x) = g(x). In addition T−g(x) := ug(0, x) can be written asT−g(x) = supX∈A[0,1]{E [g(X(1))|X(0) = x]− E[∫ 10L(X(s), βX(s,X))ds|X(0) = x]}.From this expression, we can readily check that T− is a backward Kan-torovich operator (and consequently, T is a backward linear transfer).2.8.3 Transfers associated to optimally stopped stochastictransportsIn dimensions greater than one, there are many different types of martin-gales. If one chooses those that essentially follow a Brownian path, then wehave the following examples of linear transfers.Example 2.8.9 (Optimal subharmonic martingale transfers (Ghoussoub-Kim–Palmer [31]) ). Let O be a convex bounded domain in Rd. If (µ, ν) are in49subharmonic order, i.e. µ ≺SH ν, where SH is the cone of subharmonicfunctions on O, we set,Pc(µ, ν) = infpi∈BM(µ,ν)∫O×Oc(x, y)pi(dx, dy),where each pi ∈ BM(µ, ν) is a probability measure on O×O with marginalsµ and ν, satisfying δx ≺SH pix for µ−a.e. x, where pix is the disintegrationof dpi(x, y) = dpix(y)dµ(x). Otherwise, set Pc(µ, ν) = +∞.By a theorem of Skorokhod [53], such transport plans pi can be seen asjoint distributions of (B0, Bτ ) ∼ pi, where B0 ∼ µ, Bτ ∼ ν and τ is a possiblyrandomized stopping time for the Brownian filtration. See for example [30].The above problem associated to a cost c can then be formulated asPc(µ, ν) = infτ{E[c(B0, Bτ )]; B0 ∼ µ, Bτ ∼ ν},where (Bt)t is Brownian motion starting with distribution µ and ending ata stopping time τ such that Bτ realises the distribution ν.In [31] it is shown that Pc is a backward linear transfer with a backwardKantorovich operator given by T−g(x) = Jg(x, x), whereJg(x, y) = supτ≤τOE[g(Byτ )− c(x,Byτ )],and τO is the first exit time of the set O. Under some regularity assumptionson g and c, and for each fixed x ∈ O, the function y 7→ Jg(x, y) is the uniqueviscosity solution to the obstacle problem for u ∈ C(O):u(y) ≥ g(y)− c(x, y), for y ∈ O,u(y) = g(y)− c(x, y) for y ∈ ∂O,∆u(y) ≤ 0 for y ∈ O,∆u(y) = 0 whenever u(y) > g(y)− c(x, y),50as well as the unique minimiser of the variational probleminf{∫O∣∣∇u∣∣2dy; u ≥ g − c(x, ·), u ∈ H1(O)}.Example 2.8.10 (Optimally stopped stochastic transport [28, 31]). Givena Lagrangian L : [0, 1]×Rd×Rd → R, consider the optimal stopping problemTL(µ, ν) = inf{E[∫ τ0L(t,X(t), βX(t,X(t))) dt];X(0) ∼ µ, τ ∈ S,Xτ ∼ ν,X(·) ∈ A},where S is the set of possibly randomized stopping times, and A is the classof processes defined in Section 2.8.2. In this case, TL is a backward lineartransfer with Kantorovich operator given by T−L f = Vˆf (0, ·), whereVˆf (t, x) = supX∈AsupT∈S{E[f(X(T ))−∫ TtL(s,X(s), βX(s,X)) ds∣∣∣∣X(t) = x]} ,which is, at least formally, a solution Vˆf (t, x) of the quasi-variational Hamilton-Jacobi-Bellman inequality,min{Vf (t, x)− f(x),−∂tVf (t, x)−H(t, x,∇Vf (t, x))− 12∆Vf (t, x) } = 0.2.9 Convex transfersWe saw in Example 2.5.3 that the relative entropy functional H(µ, ν) onP(X)× P(X), which we recall is defined viaH(µ, ν) :=∫Xdνdµ log(dνdµ)dµ if ν µ,+∞ otherwise,satisfies H∗µ(g) = log∫X egdµ and thus is not a linear transfer. We saw inProposition 2.5.4 that we can also express it asH∗µ(g) = inft∈R∫XTt(g)dµ, Ttg := eg+t−1 − t. (2.23)51This means thatH(µ, ν) = supg∈C(X)supt∈R{∫Xgdν −∫XTtgdµ}= supt∈Rsupg∈C(X){∫Xgdν −∫XTtgdµ}= supt∈RTt(µ, ν) (2.24)where Tt(µ, ν) := supg∈C(X){∫X gdν −∫X Ttgdµ} is a backward linear cou-pling, but not a backward linear transfer. The expression (2.24) motivatesthe following definition.Definition 2.9.1 (Convex coupling). Let T : P(X)×P(Y )→ R∪{+∞} bea proper, bounded below, weak∗ lower semi-continuous, and jointly convexfunctional. We say T is a backward convex coupling provided it is thesupremum of a family of backward linear couplings,T (µ, ν) = supi∈ITi(µ, ν)where I is some index set and for each i, Ti(µ, ν) := supg∈C(Y ){∫Y gdν −∫X T−i gdµ} is a backward linear coupling.In the same way as for forward linear couplings, we shall define T asa forward convex coupling if T˜ (ν, µ) := T (µ, ν) is a backward convexcoupling on P(Y )× P(X).We shall refer to {T−i }i∈I as the family of operators associated to T .Note they are not in general Kantorovich operators as per Definition 2.4.1.Recalling from Remark (2.4.7) that a backward linear coupling T hasT ∗µ (g) ≤∫X T−gdµ, the backward linear transfers were defined as thoselinear couplings where we have equality. In an analogous way, we see thatin general, a convex coupling T satisfies T ∗µ (g) ≤ infi∈I∫X T−i gdµ. Thismotivates the next definition.Definition 2.9.2 (Convex transfer). Let T be a backward convex coupling.We say T is a backward convex transfer if for all µ ∈ D1(T ), for all52g ∈ C(Y ), it holds thatT ∗µ (g) = infi∈I∫XT−i gdµ.Note that when I is a singleton, then T is a backward linear transfer,so all linear transfers are convex transfers. A first example of a backwardconvex transfer is of course the relative entropy via (2.23).Also note that if {δx ; x ∈ X} ⊂ D1(T ), then T ∗δx(g) = infi∈I T−i g(x),which implies that g 7→ infi∈I T−i g is a backward Kantorovich operator.More generally, for backward convex couplings, there is the inequalityT ∗µ (g) ≤∫Xinfi∈IT−i gdµ,thus T (µ, ν) is bounded above by the backward linear couplingsupg∈C(Y ){∫Ygdν −∫Xinfi∈IT−i gdµ}. (2.25)If it happens that g 7→ infi∈I T−i g is a backward Kantorovich operator (whichin general it is not, but guaranteed if D1(T ) contains the Dirac masses), thenthis linear coupling (2.25) should in fact be the linear transfer envelope T ofT provided by Proposition 2.7.4. The next proposition provides the detailsfor this.Proposition 2.9.3. Let T be a convex coupling on P(X) × P(Y ) of theformT (µ, ν) := supi∈ITi(µ, ν)where for each i ∈ I,Ti(µ, ν) = supg∈C(Y ){∫Yfdν −∫XT−i gdµ}for some map T−i : C(Y ) → USCb(X). Assume {δx;x ∈ X} ⊂ D1(T ) andsupx∈X infν∈P(Y ) T (δx, ν) < +∞. Consider the weak transport (or “trans-fer envelope”) T of T provided by Proposition 2.7.4 and the corresponding53Kantorovich operator T−. Then,1. T−is given by the formulaT−g(x) = supσ∈P(Y )infh∈C(Y ){∫Y(g − h) dσ + infiT−i h(x)}.and therefore satisfies T−g ≤ infi T−i g on C(Y ).2. If in addition each T−i is a backward Kantorovich operator, then T−g =infi∈I T−i g if and only if g 7→ infi∈IT−i g(x) is convex and lower semi-continuous.Proof. 1. We show that the hypotheses of Proposition 2.7.7 are satisfied.We have that the map g 7→ Tg := infi∈I T−i g takes C(Y ) to USCb(X), and,supx∈Xinfν∈P(Y )supg∈C(Y ){∫Ygdν − Tg(x)} = supx∈Xinfν∈P(Y )T (δx, ν) < +∞by assumption. Therefore the map Tg = infi∈I T−i g satisfies the hypothesesof Proposition 2.7.7, and we conclude that the operator defined byT−g(x) = supσ∈P(Y )infh∈C(Y ){∫Y(g − h)dσ + infi∈IT−i g(x)}is the Kantorovich operator of the weak transport T associated to T , andsatisfies T−g ≤ infi∈I T−i g.2. Suppose each T−i is a backward Kantorovich operator. If g →infi∈IT−i g(x) is convex and lower semi-continuous for any x ∈ X, then it is it-self a backward Kantorovich operator. Therefore the additional consequenceof Proposition 2.7.7 implies infi∈IT−i g = T−g.Example 2.9.4 (Linear transfer envelope of relative entropy). Recall fromthe beginning of this section or Example 2.5.3, thatH∗µ(g) = inft∈R∫X Tt(g)dµ,where Ttg := eg+t−1 − t. It is easy to compute thatinft∈R{eg(x)+t−1 − t} = g(x)54and consequently T−g = g is the Kantorovich operator for the linear transferenvelope. Therefore T is nothing but the trivial linear backward transfer ofExample 2.5.7T (µ, ν) =0 if µ = ν,+∞ otherwise.A natural question to investigate is whether convex functions of back-ward linear transfers, are backward convex transfers. The next propositionsays this is indeed the case.Proposition 2.9.5. Let α : [0,∞)→ R be a convex increasing function, andT a backward linear transfer with corresponding operator T−. Then α(T ) isa backward convex transfer, and its envelope α(T ) as given in Proposition2.7.4 has Kantorovich operator given byT−α g = infs>0{sT−(gs) + α⊕(s)}, (2.26)where α⊕(t) := sup{ts−α(s); s ≥ 0} corresponds to the Legendre transformof the extended real-valued function α˜(t) :=α(t) t ≥ 0,+∞ otherwise.Proof. Note that α˜ is convex and lower semi-continuous, hence α˜∗∗(t) = α˜(t)and so α(t) = sup{ts− α⊕(s); s ≥ 0}. Therefore,α(T (µ, ν)) = sups≥0{sT (µ, ν)− α⊕(s)}= sups≥0supg∈C(Y ){∫Ygdν −∫X(sT−(gs) + α⊕(s))dµ}so we see that α(T ) is a backward convex coupling.To show that α(T ) is a backward convex transfer, we have that ν 7→α(T (µ, ν)) is convex and weak∗ lower semi-continuous, thus we just need to55compute the Legendre transform for µ ∈ D1(T ),α(T )∗µ(g) = supν∈P(Y ){∫Ygdν − α(T (µ, ν))},where we recall the notation α(T )∗µ is the Legendre transform of ν 7→α(T (µ, ν)). Substituting α(t) = sup{ts−α⊕(s); s ≥ 0} for α(T (µ, ν)) in theabove expression, we obtainα(T )∗µ(g) = supν∈P(Y )infs≥0{∫Ygdν − sT (µ, ν) + α⊕(s)}= supν: T (µ,ν)<+∞infs≥0{∫Ygdν − sT (µ, ν) + α⊕(s)}Notice now that ν 7→ f(ν, s) := ∫Y gdν − sT (µ, ν) + α⊕(s), as a functionon the convex and weak∗ closed subset V := {ν ∈ P(Y ) ; T (µ, ν) < +∞},is weak∗ upper semi-continuous, and also quasi-concave: {ν ; f(ν, s) ≥ λ} isconvex or empty for λ ∈ R. At the same time, s 7→ f(ν, s) on W := [0,∞)is lower semi-continuous on W and also quasi-convex: {s ; f(ν, s) ≤ λ} isconvex or empty for λ ∈ R. Then by Sion’s minimax theorem (Theorem2.7.6), we may interchange sup and inf to obtainα(T )∗µ(g) = infs≥0supν∈P(Y ){∫Ygdν − sT (µ, ν) + α⊕(s)}= infs≥0∫Y[sT−(gs) + α⊕(s)]dµ.where we have used the fact that sT is a backward linear transfer withoperator g 7→ sT−(gs ) (see Section 2.10).Regarding the envelope α(T ) we note that for each s > 0, T−s g :=sT−(gs ) + α⊕(s) is a backward Kantorovich operator, and, moreover, thefunction (s, g) 7→ sT−(gs ) + α⊕(s) is jointly convex on R+ × C(Y ). Hence,the infimum in s is convex, and therefore g 7→ infs≥0{sT−(gs ) + α⊕(s)} isconvex. We conclude by Proposition 2.9.3.Example 2.9.6 (Generalised entropy functional). Consider for a strictly56convex function α : [0,∞) → R, the generalised entropy functional onP(X)× P(X), given by,Tα(µ, ν) =∫X α(dνdµ)dµ if ν µ,+∞ otherwise.By a similar argument as given in Example 2.5.3 (see [35], Proposition 2.9),we have(Tα)∗µ(g) = inft∈R∫X[α⊕(t+ g(x))− t]dµ(x)where α⊕ is as in Proposition 2.9.5, so that Tα is a backward convex transfer.Example 2.9.7 (A backward convex coupling which is not a convex trans-fer). Let Ω ⊂ Rd be compact with 1 < |Ω| <∞, λ := 1|Ω| , and define for anytwo given probability measures µ, ν on Ω,Tλ(µ, ν) =0 if λ dνdµ ≤ 1µ-a.e.+∞ otherwise,Note that when µ = λdx|Ω (the uniform measure on Ω),Tλ(λdx|Ω, ν) =0 if dνdx ≤ 1 Lebesgue-a.e.+∞ otherwise.We claim that Tλ is a backward convex coupling but not a convex transfer.Indeed, for the first claim, consider αm,λ(t) := (λt)m log(λt) for m ≥ 1 andt ≥ 0, and defineTm,λ(µ, ν) :=∫Ω αm,λ(dνdµ)dµ, if ν << µ,+∞ otherwise.By Proposition 2.9.5, Tm,λ is a backward convex transfer and for g ∈ C(Ω),(Tm,λ)∗µ(g) = inf{∫Ω[α⊕m,λ(g(x) + t)− t]dµ(x) ; t ∈ R}57where we recall (Tm,λ)∗µ is the Legendre transform of the map ν 7→ Tm,λ(µ, ν).The function α⊕m,λ can be explicitly computed asα⊕m,λ(t) =e−1+ 1m−1W (βmt)[βmt+1meW (βmt)]if t ≥ − λm−1e−1,0 if t < − λm−1e−1.where βm :=m−1λm em−1m , and W is the Lambert-W function (the multi-valuedinverse of w 7→ wew). It is easy to see that Tλ(µ, ν) = supm Tm,λ(µ, ν);hence it is a backward convex coupling (as a supremum of backward convextransfers).However, Tλ is not a backward convex transfer itself, since(Tλ)∗µ(g) = (supmTm,λ)∗µ(g) ≤ infm (Tm,λ)∗µ(g) =∫Ωgλdµ,with the inequality being in general strict.One well-studied problem for which this is relevant is the quadraticWasserstein projection. Denoting W 22 (µ, ν) as the optimal transport withcost function c(x, y) = |x − y|2, the quadratic Wasserstein projection of ameasure ν onto the set of Lebesgue densities bounded above by 1, is theminimiser of the following variational problem,inf{W 22 (σ, ν) ; σ ∈ P(Ω),dσdx≤ 1}.This problem has received a lot of attention in the context of congestedcrowd motion (see, e.g. [49]). By writinginf{W 22 (σ, ν);dσdx≤ 1} = inf{Tλ(λdx|Ω, σ) +W 22 (σ, ν) ; σ ∈ P(Ω)}we see that it consists of an inf-convolution of the backward convex cou-pling Tλ with the backward linear transfer W 22 . Therefore we cannot takeadvantage of the inf-convolution property for backward convex transfers andbackward linear transfers (see Section 2.10.2) to write a dual formulation.582.9.1 Backward entropic transfersWe saw previously that the relative entropy functional was a backwardconvex transfer. However, there is further structure: not only H∗µ(g) =inft∈R∫Y T−t gdµ (see (2.23)), but in fact H∗µ(g) = log(∫Y egdµ). We thusisolate a special subclass of backward convex transfers.Definition 2.9.8. Let β : R → R be concave and increasing, and let T :P(X) × P(Y ) → R ∪ {+∞} be a bounded below, proper, weak∗ lowersemi-continuous and convex functional. We say that T is a backwardβ-entropic transfer if there exists a map T− : C(Y ) → USC(X) suchthat for every µ ∈ D1(T ),T ∗µ (g) = β(∫XT−gdµ), for any g ∈ C(Y ).We note from the definition that a backward β-entropic transfer is abackward convex transfer. This follows from the concavity of β, so thatβ(t) = infs∈R{−st + (−β)∗(s)} where (−β)∗ is the Legendre transform of−β.2.10 Operations on convex and linear transfersIn this section, we highlight a few basic operations on convex transfers. Inparticular, the notion of inf-convolution (see below in Section 2.10.2) willbe, in subsequent chapters, the operation which is of most interest.2.10.1 Linear transfers are a convex coneIn this section, we highlight some basic operations for convex and lin-ear transfers, the most important for our purposes in Chapter 5, is inf-convolution.Proposition 2.10.1. 1. (Scalar multiplication for convex trans-fers) If a ∈ R+ \ {0} and T is a backward convex transfer, then (aT )given by59(aT )(µ, ν) := aT (µ, ν)is also a backward convex transfer with(aT )∗µ(g) = infi∈I∫XaT−i (ga)dµ. (2.27)2. (Addition for linear transfers) If T1 and T2 are backward lineartransfers on X × Y such that {δx ; x ∈ X} ⊂ D(T1) ∩ D(T2) withKantorovich operator T−1 , T−2 respectively, then the sum defined as(T1 ⊕ T2)(µ, ν) := inf{∫X{T1(x, pix) + T2(x, pix)} dµ(x);pi ∈ K(µ, ν)}is a backward linear transfer on X × Y , with Kantorovich operatorgiven on C(Y ) byT−g(x) = supσ∈P(Y ){∫Yg dσ − T1(δx, σ)− T2(δx, σ)}= infh∈C(Y ){T−1 h(x) + T−2 (g − h)(x)}.2.10.2 Inf-convolutionAn important operation from the perspective of ergodic theory, is the notionof inf-convolution. We will see shortly that this will allow us to iteratea linear transfer which we shall investigate and go into greater detail inChapter 5.Definition 2.10.2 (Inf-convolution). Let X1, X2, X3 be 3 compact, com-plete, and separable metric spaces, and suppose T1 (resp., T2) are functionalson P(X1)×P(X2) (resp., P(X2)×P(X3)). The inf-convolution of T1 andT2 is the functional on P(X1)× P(X3) given byT (µ, ν) := T1 ? T2 = inf{T1(µ, σ) + T2(σ, ν); σ ∈ P(X2)}.60By induction, one can then consider inf-convolution of n functionals inthe obvious way. Note that inf-convolution preserves convexity and weak∗lower semi-continuity. This leads to the following important stability prop-erty.Proposition 2.10.3. If T is a backward linear transfer on P(Y ) × P(Z)with Kantorovich operator T− : C(Z)→ C(Y ), and F is a backward convextransfer on P(X) × P(Y ), then F ? T is a backward convex transfer onP(X)× P(Z), with(F ? T )∗µ(h) = infi∈I∫XF−i ◦ T−h dµ.Proof. We calculate the Legendre dual of (F ? T )µ at g ∈ C(Z) and obtain,(F ? T )∗µ(g) = supν∈P(Z)supσ∈P(Y ){∫Zg dν −F(µ, σ)− T (σ, ν)}= supσ∈P(Y ){T ∗σ (g)−F(µ, σ)}= supσ∈P(Y ){∫YT−g dσ −F(µ, σ)}= (F)∗µ(T−(g))= infi∈I∫XF−i ◦ T−g(x)) dµ(x).Corollary 2.10.4. If T1 (resp., T2) is a backward linear transfer on P(X1)×P(X2) (resp., on P(X2)×P(X3)) with Kantorovich operator T−1 : C(X2)→USC(X1) (resp., T−2 : C(X3) → C(X2)), then T1 ? T2 is also a backwardlinear transfer on P(X1)× P(X3) with Kantorovich operator equal to T−1 ◦T−2 .Remark 2.10.5. The assumption that T−2 maps continuous functions tocontinuous functions is too restrictive; the appropriate setting is for uppersemi-continuous functions, but strictly speaking in that case T−1 ◦ T−2 g isnot defined. This is a non-issue, but requires us to spend some words in61Chapter 5 on extension of the operators to upper semi-continuous functions,in which case the above proposition still holds in the more general case whenT−2 : C(X3)→ USC(X2)By induction on the convolution property enjoyed by linear transfers,one can immediately show the following.Corollary 2.10.6. Let X0, X1, ...., Xn be (n+ 1) compact spaces, and sup-pose for each i = 1, ..., n, we have functionals Ti on P(Xi−1)× P(Xi). Forany probability measures µ on X0 and ν on Xn, defineT (µ, ν) := T1 ? T2 ? . . . ? Tn(µ, ν)= inf{T1(µ, ν1) + T2(ν1, ν2)...+ Tn(νn−1, ν); νi ∈ P(Xi), i = 1, ..., n− 1}.If each Ti is a backward (resp., forward) linear transfer with a correspond-ing Kantorovich operator T−i : C(Xi) → C(Xi−1) (resp., T+i : C(Xi) →C(Xi+1)), then T is a backward (resp., forward) linear transfer with a Kan-torovich operator given byT− = T−1 ◦ T−2 ◦ ... ◦ T−n (resp., T+ = T+n ◦ T+n−1 ◦ ... ◦ T+1 )In other words, the following duality formula holds:T (µ, ν) = sup{∫Xng dν −∫X0T−1 ◦ T−2 ◦ ... ◦ T−n gdµ; g ∈ C(Xn)}.respectively,T (µ, ν) = sup{∫XnT+n ◦ T+n−1 ◦ ...T+1 f dν −∫X0f dµ; f ∈ C(X0)}We will study in Chapter 5, iterates of a single linear transfer T ? T ?. . . ? T .2.10.3 Tensor products and dual sums of linear transfersWe highlight other operations for linear transfers.62Definition 2.10.7. 1. (Tensor product) If T1 (resp., T2) are functionalson P(X1)×P(Y1) (resp., P(X2)×P(Y2)) such that {δx1 ; x1 ∈ X1} ⊂D(T1) and {δx2 ; x2 ∈ X2} ⊂ D(T2), then the tensor product of T1and T2 is the functional on P(X1 ×X2)× P(Y1 × Y2) defined by:T1 ⊗ T2(µ, ν) := inf{∫X1×X2(T1(δx1 , pi1x1,x2) + T2(δx2 , pi2x1,x2)) dµ(x1, x2);pi ∈ K(µ, ν)}where dpi(x1, x2, y1, y2) = dpix1x2(y1, y2)dµ(x1, x2), and piix1x2 = ProjYi#pix1x2is the projection of pix1x2 onto Yi, i = 1, 2.2. (Dual sum) If T1 and T2 are backward linear transfers on P(X)×P(Y )such that {δx ; x ∈ X} ⊂ D1(T1)∩D1(T2), with Kantorovich operatorT−1 , T−2 respectively, then the dual sum, T1T2, is defined as thetransfer whose Kantorovich operator is T−1 + T−2 , that isT1T2(µ, ν) := sup{∫Ygdν −∫X(T−1 g + T−2 g)dµ; g ∈ C(Y )}The definition of tensorization is via a weak transport, so it is not sur-prising that we have the following stability property.Proposition 2.10.8. If T1 (resp., T2) is a backward linear transfer onP(X1) × P(Y1) (resp., P(X2) × P(Y2)) such that {δx ; x ∈ X1} ⊂ D1(T1)and {δx ; x ∈ X2} ⊂ D1(T2), then T1 ⊗ T2 is a backward linear transfer onP(X1 ×X2)× P(Y1 × Y2), with Kantorovich operator given byT−g(x1, x2) = supσ∈K(σ1,σ2){∫Y1×Y2gdσ − T1(δx1 , σ1)− T2(δx2 , σ2)}. (2.28)Moreover,T1 ⊗ T2(µ, ν1 ⊗ ν2) ≤ T1(µ1, ν1) +∫X2T2(δx2 , ν2) dµ2(x2), (2.29)where µ2 = ProjX2#µ.Proof. The tensor product as defined above is a weak transport, where the63cost on X1 ×X2 × P(Y1 × Y2) is simply,c((x1, x2), σ) := T1(δx1 , σ1) + T2(δx2 , σ2),where σi = projYi#σ, i = 1, 2, are the marginals of σ on Y1 and Y2 re-spectively. The cost c is bounded below, lower semi-contiuous, and σ 7→c((x1, x2), σ) is convex. Therefore by Theorem 2.7.2, T1 ⊗ T2 is a backwardlinear transfer withT−g(x1, x2) := supσ∈P(Y1×Y2){∫Y1×Y2gdσ − c((x1, x2), σ)}.For the upper bound (2.29) for T1⊗T2, since pi = µ⊗ (ν1⊗ ν2) is admissiblein the infimum, we haveT1 ⊗ T2(µ, ν1 ⊗ ν2) ≤ T1(µ1, ν1) +∫X1×X2T2(δx2 , ν2) dµ(x1, x2)2.10.4 Projections and Hopf-Lax formulaeWe now investigate linear transfers which arise via inf-convolution.Example 2.10.9 (Projection onto the set of balaye´es of a given measure).Let T be a backward linear transfer on P(X) × P(Y ) and consider thefollowing “projection” problemS(µ, ν) := inf{T (µ, σ);σ ∈ Kν}, (2.30)where Kν is a closed convex subset of P(Y ) that depends on ν. Define onP(Y )× P(Y ),I(σ, ν) :=0 if σ ∈ Kν+∞ otherwise.64so that we may write S asS(µ, ν) = T ? I(µ, ν) = infσ∈P(Y ){T (µ, σ) + I(σ, ν)}.In general, S will not be a backward linear transfer. However, consider thecase when Kν = {σ ; σ A ν} is the set of probability measures in A-orderwith ν, for a convex cone A of continuous functions (see Section 2.5.3).Then I = B where B is the balayage of measures from Section 2.5.3, andthe inf-convolution property of Section 2.10.2 implies that S is a backwardlinear transfer with backward Kantorovich operatorS−g(x) := T−(gˆ)(x),where we recall, again from Section 2.5.3, thatgˆ(x) = inf{h(x) ; h ∈ −A, h ≥ g}.A particular setting that has been studied in, e.g. [33], [22], is when T isthe optimal transport with quadratic cost c(x, y) = |x − y|2 and A is thecollection of convex functions on some compact subset of Rn. In that case,we have seen from Section 2.5.3 that gˆ is the concave envelope of g, hencewe obtain the dualityinf{T (µ, σ);σ A ν} = supg∈C(X){∫Xgˆdν −∫XT−gdµ}where T−g(x) = supy∈X{g(y) − |x − y|2} is the Kantorovich operator ofoptimal transport (see Section 2.6). By noting gˆ ≥ g, it follows that T−(gˆ) ≥T−(g), so it actually suffices to restrict the supremum to g which are concave,in which case gˆ = g. The conclusion isinf{T (µ, σ);σ A ν} = supg concave{∫Xgdν −∫XT−gdµ}which is the duality obtained by Gozlan et. al. (see [36], Theorem 2.11 (3)).65It is easy to see that the convolution of two optimal mass transportswith costs c1 and c2 respectively, is also a mass transport corresponding toa cost functional given by the convolution c1?c2. We use the inf-convolutionproperty of linear transfers to establish this easy result.Proposition 2.10.10 (Lifting convolutions to Wasserstein space). Let Xi,i = 0, 1, 2, be compact spaces, and, ci : Xi × Xi+1 → R ∪ {+∞}, i =0, 1, lower semi-continuous cost functions. Define the corresponding optimalmass transportsTci(µ, ν) = infpi∈K(µ,ν){∫Xi×Xi+1ci(x, y) dpi(x, y)},with corresponding forward and backward Kantorovich operators T+ci andT−ci as in Section 2.6. Consider the cost function on X0 ×X2 which is theconvolution of the two cost functions,c(x, x′) := c1 ? c2(x, x′) = inf{c1(x, x1) + c2(x1, x′); xi ∈ Xi}.ThenTc(µ, ν) = Tc1 ? Tc2(µ, ν)= supf∈C(X0){∫X2T+c1 ◦ T+c2f dν −∫X0f dµ}= supg∈C(X2){∫X2g dν −∫X0T−c1 ◦ T−c2gdµ}.Proof. By the inf-convolution property of Section 2.10.2, Tc1?c2 has backward66Kantorovich operator equal toT−c1 ◦ T−c2g(x) = supx1∈X1{T−c2g(x1)− c1(x, x1)}= supx1∈X1,x2∈X2{g(x2)− c2(x1, x2)− c1(x, x1)}= supx2∈X2{g(x2)− c(x, x2)}= T−c g(x).The next two problems are technically defined on Rd and not on a com-pact space, so strictly speaking, they do not fit into our compact framework.One then needs to change to the spaceM1(Rd) of finite Borel measures withfinite first moments, in duality with Lip(M) the bounded uniformly Lips-chitz functions on M .Example 2.10.11 (The ballistic transfer (Barton-Ghoussoub [4])). Let Lbe a Tonelli Lagrangian, which we do not make precise here but insteaddefine later in Chapter 4, which satisfies appropriate assumptions (see [4],(A0)). For M = Rd = M∗ (M∗ the dual space), the deterministic ballisticmass transport is defined asB(µ, ν) := inf{∫M∗×Mb(v, x)dpi(v, x) ; pi ∈ K(µ, ν)},whereb(v, x) := inf{〈v, γ(0)〉+∫ 10L(t, γ(t), γ˙(t)) dt; γ ∈ C1([0, 1),M), γ(1) = x}.Note that we can write b in the formb(v, x) = inf{〈v, y〉+ cL(y, x); y ∈M},where the cost cL is given by the generating function associated to L inExample 2.6.6, which means that b is the convolution of the Brenier cost of67Example 2.6.7 with the cost induced by the Lagrangian L.By the inf-convolution property 2.10.3, it is both a forward, and back-ward, linear transfer. In particular, its backward Kantorovich operator isthen the composition of the two corresponding backward Kantorovich oper-ators, and we concludeB(µ, ν) = sup{∫Mg(x)dν(x)−∫M∗(−Wg(0, ·))∗(−v)dν(v) ; g ∈ Lip(M)}whereWg solves the Hamilton-Jacobi equation of Example 2.6.6 withWg(1, x) =g(x), and where h∗(v) := supx∈M{〈v, x〉 − h(x)} is the Legendre transformof a function h on M (see Theorem 2 in [4]).Example 2.10.12 (Stochastic ballistic transfer (Barton-Ghoussoub [4])).For the stochastic ballistic transport problem,Bs(µ, ν) := inf{E[〈V,X(0)〉+∫ 10L(t,X, βX(t,X)) dt]; V ∼ µ, X(·) ∈ A[0,1], X(1) ∼ ν},where we are using the same notation for the stochastic processes as inSection 2.8.2, this is a convolution of the Brenier-Wasserstein transfer ofExample 2.6.7 with now the stochastic transport of Example 2.8.7. Undersuitable conditions on L (see [4], (A1), (A2), (A3)), we have, similarly tothe deterministic version,Bs(µ, ν) = sup{∫Mg(x) dν(x)−∫M∗(−ψg(0, ·))∗(−v) dµ(v); g ∈ Lip(M)},only now ψg is the solution to the Hamilton-Jacobi-Bellman equation∂ψ∂t+12∆ψ(t, x) +H(t, x,∇ψ) = 0, ψ(1, x) = g(x).In other words, Bs is a backward linear transform with Kantorovich operatorT−g(v) = (−ψg(0, ·))∗(−v).682.11 Subdifferentials of convex transfersIf T is a backward linear transfer, then ν 7→ T (µ, ν) =: Tµ(ν) is convexand weak∗ lower semi-continuous. Therefore one can consider its (weak∗)subdifferential ∂Tµ given byg ∈ ∂Tµ(ν) if and only if T (µ, ν ′) ≥ T (µ, ν) +∫Y gd(ν′ − ν) for any ν ′ ∈ P(Y ).In other words, g ∈ ∂Tµ(ν) if and only if Tµ(ν) + T ∗µ (g) = 〈g, ν〉. SinceTµ(ν) = T (µ, ν) and T ∗µ (g) =∫T−gdµ, we then obtain the following charac-terization of the subdifferentials, which says that the subdifferential ∂Tµ(ν)is simply all those g for which dual attainment holds for T (µ, ν).Proposition 2.11.1. Let T be a backward linear transfer. For any µ ∈D1(T ), the subdifferential of Tµ : P(Y ) → R ∪ {+∞} at ν ∈ P(Y ) is givenby∂Tµ(ν) ={g ∈ C(Y ) :∫Yg(y)dν(y)−∫XT−g(x)dµ(x) = T (µ, ν)}It is easy to see that the same expressions hold (with the necessarymodifications) for backward convex transfers.In the following, we observe some elementary consequences for elementsin the subdifferential.Proposition 2.11.2. Suppose T is a linear backward transfer such that{δx ; x ∈ X} ⊂ D1(T ). Fix µ ∈ P(X) and ν ∈ P(Y ) such that T (µ, ν) <+∞. Then, there exists p¯i ∈ K(µ, ν) such that for each g¯ ∈ ∂Tµ(ν), we haveT−g¯(x) =∫Yg¯(y)dp¯ix(y)− T (δx, p¯ix), for µ-a.e. x ∈ X,where p¯ix is a disintegration of p¯i with w.r.t. µ.Conversely, if ν 7→ T (µ, ν) is strictly convex and g¯ ∈ ∂Tµ(ν) for someν ∈ P(Y ). If x→ σx is any Borel-measurable selection such thatT−g¯(x) = supσ{∫g¯dσ − T (δx, σ)}=∫Yg¯dσx − T (δx, σx),69then T (µ, ν) is attained by the measure p¯i defined via dp¯i(x, y) = dσx(y)dµ(x).Proof. By a recent result [3], there exists p¯i ∈ K(µ, ν) such thatT (µ, ν) =∫XT (δx, p¯ix)dµ(x).If f¯ ∈ ∂Tµ(ν), then by definition∫Yf¯(y)dν(y)−∫XT−f¯(x)dµ(x) = T (µ, ν) =∫XT (x, p¯ix)dµ(x),that is∫X[T−f¯(x)− ∫Y f¯(y)dp¯ix(y) + T (x, p¯ix)] dµ = 0. Since T−f¯(x) =supσ{∫f¯dσ − T (x, σ)}, the quantity in the brackets is non-negative andwe get our claim.Conversely, If f¯ ∈ ∂Tµ(ν) is non-empty for some ν ∈ P(Y ), then∫f¯dν−∫T−f¯dµ = T (µ, ν). From the expression T−f¯(x) = supσ{∫Y f¯dσ − T (δx, σ)},we know the supremum will be achieved by some σx. Defining p˜i by dp˜i(x, y) =dµ(x)dσx(y), and the right marginal of p˜i by ν˜, we integrate against µ toachieve ∫T−f¯dµ =∫f¯dν˜ −∫T (δx, σx)dµ.This shows that T (µ, ν˜) = infpi∈K(µ,ν˜)∫ T (δx, pix)dµ = ∫ T (δx, σx)dµ, andconsequently, f¯ ∈ ∂Tµ(ν˜). But by strict convexity, this can only be true ifν˜ = ν.Note that while we can use general existence results such as the Brondsted-Rockafellar theorem [46], to state that ∂Tµ(ν) exists for a weak∗-dense set ofν ∈ P(Y ) (and therefore the dual problem is “generically” attained for thisweak∗-dense set), proving attainment in the dual problem is, in general, adifficult problem.Corollary 2.11.3. Suppose T is a backward linear transfer on P(X)×P(Y )such that the Dirac masses are in D1(T ), and fix µ ∈ P(X). Then for everyν ∈ P(Y ) and every  > 0, there exists ν ∈ P(Y ) such that W2(ν, ν) < and the dual problem for T (µ, ν) is attained.70We briefly mention the following Euler-Lagrange equation for variationalproblems on spaces of measures, which follows closely [24] (see in particularTheorem 2.2 there).Proposition 2.11.4. Let Tα(µ, ν) :=∫X α(dνdµ)dµ be the generalised en-tropy functional considered in Example 2.9.6, where α : [0,∞) → R is inC1([0,∞)), strictly convex, and superlinear. Let T be any backward lineartransfer. For a fixed µ, consider the functional Iµ(ν) := Tα(µ, ν)− T (µ, ν),and assume ν¯ realises infν∈P(X) Iµ(ν). Then, there exists f¯ ∈ ∂Tµ(ν¯) suchthat the following Euler-Lagrange equation holds for ν¯−a.e. x ∈ X,α′(dν¯dµ)= f¯ + C,where C is a constant.Proof. Recall from Example 2.9.6 that Tα is a backward convex transferwithT ∗µ (g) = inft∈R{∫X[α⊕(g(x) + t)− t]dµ(x)},where α⊕(t) := sups≥0{st− α(t)}. It follows thatα′(dνdµ)∈ ∂Tµ(ν).We can see this either directly from the subdifferential definition, or fromobservingα⊕(α′(dνdµ))=dνdµα′(dνdµ)− α(dνdµ).In particular,T ∗µ(α′(dνdµ))=∫Xα⊕(α′(dνdµ))dµ.The rest is a straightforward adaptation of Theorem 2.2 in [24]..71Chapter 3Dualities for transferinequalities3.1 IntroductionLet Tc be an optimal transport associated to a cost function c on X × X,and let J be a functional on P(X) × P(X). A family of inequalities, thetransport inequalities, compare the cost Tc(µ, ν) of transporting a measureν ∈ P(X) to a fixed “reference measure” µ ∈ P(X), in the form (see, e.g.the survey article [34])α(Tc(ν, µ)) ≤ J(µ, ν) or α(Tc(µ, ν)) ≤ J(µ, ν),where α : [0,∞) → [0,∞) is a given increasing function with α(0) = 0. Aclassical choice for J is the relative entropy functionalH(µ, ν) = ∫X log( dνdµ) dν if ν µ and +∞ otherwise,in which case they are known as the transport-entropy inequalities, includingTalagrand’s transport inequality for the Gaussian measure [57]. One interestof transport-entropy inequalities is that they imply concentration of measurephenomena (see e.g. Theorem 1.7 in [34] originally due to Marton [41, 42],72or the survey [38]).In order to prove a transport-entropy inequality, one can take advantageof the fact that both Tc and H have dualities, which in the present context,is to exploit the fact that Tc is a linear transfer, and H is a convex transfer.As was originally proved by Bobkov and Go¨tze [8], one can show (see [34])the following equivalence.Theorem 3.1.1 ([34], Theorem 3.2, Corollary 3.3). Let c be a lower semi-continuous cost function, α : [0,∞)→ [0,∞) convex, increasing, and α(0) =0, and fix a reference measure µ ∈ P(X). Then the following are equivalent:1. α(Tc(ν, µ)) ≤ H(µ, ν) for all ν ∈ P(X),2. For all g ∈ C(X),∫Xe−sT−c gdµ ≤∫Xe−sgdµ+α⊕(s), s ≥ 0where we recall that α⊕(s) := supt≥0{st−α(t)} for s ≥ 0, and T−c g(x) =supy∈X{g(y)− c(x, y)}.Proof. The proof is illustrative so we shall provide it here: We have as abackward linear transfer,Tc(ν, µ) = supg∈C(X){∫Xgdµ−∫XT−c gdν}as well as α(t) = sups≥0{st−α⊕(s)} (this follows by extending α to +∞ fort < 0 (denoting this by α˜) and noting α is convex and lower semi-continuousso α(t) = sups∈R{st− α˜∗(s)} = sups≥0{st−α⊕(s)}). Then for all g ∈ C(X)and all s ≥ 0, it follows thatα(Tc(ν, µ)) = sups≥0supg∈C(X){∫Xsgdµ−∫XsT−c gdν − α⊕(s)}.Then we writeα(Tc(ν, µ)) ≤ H(µ, ν) for all ν ∈ P(X)⇐⇒ 0 ≤ infν∈P(X){H(µ, ν)− α(Tc(ν, µ))}.73Substituting directly the above expression for α(Tc(ν, µ)) yields0 ≤ infν∈P(X)infg∈C(X)infs≥0{H(µ, ν)−∫Xsgdµ+∫XsT−c gdν + α⊕(s)}= infg∈C(X)infs≥0{∫X(−sg + α⊕(s))dµ− supν∈P(X){∫X(−T−c g)dν −H(µ, ν)}}= infg∈C(X)infs≥0{∫X(−sg + α⊕(s))dµ−H∗µ(−sT−c g)}= infg∈C(X)infs≥0{∫X(−sg + α⊕(s))dµ− log∫Xe−sT−c gdµ}.Hence by rearranging and taking the exponential, we obtain∫Xe−sT−c gdµ ≤ e∫X −sgdµ+α⊕(s) for all g ∈ C(X) and all s ≥ 0.The goal now is to deduce analogous duality statements when α(Tc) isreplaced by T which is a linear or convex coupling, and when the relativeentropy H is replaced by a entropic, or more generally, a convex transfer(recall Definition 2.9.8). The main interest for computing the dualities isthat they transform inequalities between measures, to inequalities on func-tions. Therefore if one wishes to prove that a particular transfer inequalityholds, they can instead prove the equivalent functional inequality, for whichone may hope to take advantage of the extensive literature that has beendeveloped for functional inequalities. As an example, see [8].We shall assume throughout that the operators associated to the cou-plings/transfers discussed in this chapter, map continuous functions into atleast the space of upper semi-continuous functions. In that case, we use,in advance, the inf-convolution property Corollary 5.1.5 to justify certainLegendre transform computations, which is the statement of Proposition2.10.3 and Corollary 2.10.4 extended to upper semi-continuous functions(note Chapter 5 does not depend on Chapter 3).743.2 Backward convex coupling and backwardlinear transferFix µ ∈ P(X). As a generalization of the transport-entropy inequality oftypeα(Tc(σ, µ)) ≤ H(µ, σ) for all σ,we would like to prove inequalities such as,F2(σ, µ) ≤ F1(µ, σ) for all σ ∈ P(X),where F1 is a backward convex transfer and F2 is a backward convex cou-pling. In this section, we provide the dual characterization for such an in-equality, and refine it further to the special case when F1 = λE ? T . To findthe dual characterization, we compute the duality for the infσ{F1(µ, σ) −F2(σ, ν)} where µ and ν are some fixed probability measures.Proposition 3.2.1. Let F1 be a backward convex transfer on P(X1)×P(Y1)with operators (F−1,i)i∈I , and F2 a backward convex coupling on P(Y1) ×P(X2) with operators (F−2,j)j∈J . Then for fixed µ ∈ P(X1) and ν ∈ P(X2),we haveinfσ∈P(Y1){F1(µ, σ)−F2(σ, ν)} = infg∈C(X2)infj∈J{− infi∈I∫X1F−1,i(−F−2,jg)dµ−∫X2gdν}.Proof. As F2 is a backward convex coupling, we may write F2(σ, ν) =supg∈C(X2) supj∈J{∫X2gdν−∫Y1 F−2,jgdσ}. We may then substitute this intothe expressioninfσ∈P(Y1){F1(µ, σ)−F2(σ, ν)} = infσ∈P(Y1)infg∈C(X2)infj∈J{F1(µ, σ)−∫X2gdν +∫Y1F−2,jgdσ}= infg∈C(X2)infj∈J{− supσ∈P(Y1){∫Y1(−F−2,jg)dσ −F1(µ, σ)} −∫X2gdν}= infg∈C(X2)infj∈J{− infi∈I∫X1F−1,i(−F−2,jg)dµ−∫X2gdν}where the last equality follows since F1 is a backward convex transfer and75thus has the specified Legendre transform.Corollary 3.2.2. Under the same hypotheses as the above Proposition 3.2.1,the following are equivalent:1. For all σ ∈ P(Y1), we haveF2(σ, ν) ≤ F1(µ, σ).2. For all g ∈ C(X2), it holdssupj∈Jinfi∈I∫X1F−1,i(−F−2,j(g))dµ+∫X2gdν ≤ 0.Proof. By Proposition 3.2.1 it follows immediately that0 ≤ infσ∈P(Y1){F1(µ, σ)−F2(σ, ν)} if and only if 0 ≤ infg∈C(X2)infj∈J{− infi∈I∫X1F−1,i(−F−2,jg)dµ−∫X2gdν}.This equivalence is easily seen by rearrangement to be the statement ofequivalence between Item 1 and 2 in the above statement.Corollary 3.2.3. Specializing Corollary 3.2.2 to the case where F1 := λE?T(and relabeling F2 to F) for E a backward β-entropic transfer on P(X1) ×P(Y1) with operator E−, T a backward linear transfer on P(Y1) × P(Y2)with Kantorovich operator T−, and λ > 0, the following are equivalent:1. For all σ ∈ P(Y2), we haveF(σ, ν) ≤ λ E ? T (µ, σ).2. For all g ∈ C(X2),supj∈Jβ(∫X1E− ◦ T−(−F−j (λg)λ)dµ)+∫X2g dν ≤ 0.Proof. Note that the inf-convolution λE ? T is a backward (λβ)-entropictransfer (in particular, a backward convex transfer) with operator g 7→76E− ◦T−( gλ), therefore the above proposition applies and we obtain the dualinequalitysupj∈Jλβ(∫X1E− ◦ T−(−F−j (g)λ)dµ)+∫X2g dν ≤ 0 for all g ∈ C(X2).Dividing by λ and relabeling h := gλ , the inequality has to hold for allh ∈ C(X2), which completes the equivalence.Corollary 3.2.4. Under the same setting as the above Corollary ??, butwith E = H the relative entropy, the following are equivalent.1. For all σ ∈ P(Y2), we have F(σ, ν) ≤ λH ? T (µ, σ).2. For all g ∈ C(X2),supj∈J∫X1eT−(−F−j(λg)λ)dµ ≤ e−∫X2g dν.In particular, if T is the identity transfer and F is a backward linear transfer,then the following are equivalent:1. For all σ ∈ P(Y2), we have F(σ, ν) ≤ λH (µ, σ)2. For all g ∈ C(X2), we have∫X1e−F−(λg)λ dµ ≤ e−∫X2g dν.Proof. Taking in the above Corollary E = H the relative entropy, thenβ(t) = log(t) and E−g = eg, and we deduce that for all g ∈ C(X2 and allj ∈ J ,log∫X1eT−(−F−j(λg)λ)dµ+∫X2g dν ≤ 0,which gives the asserted equivalence. If further T is the identity transfer(i.e. T (µ, ν) = 0 if µ = ν, +∞ otherwise), then T−g = g.773.3 Forward convex coupling and backwardlinear transferWe are now interested in inequalities such asF2(ν, σ) ≤ F1(µ, σ) for all σ ∈ P(X), (3.1)where F1 is a backward convex transfer and F2 is a backward convexcoupling (which, if we define F˜2(σ, ν) := F2(ν, σ), we can view as an in-equality between a forward convex coupling and backward linear transfer,hence the name of the section). As in the previous section, to compute thedual inequality to (3.1), we find the dual expression for the inf-convolutioninfσ∈P(X){F1(µ, σ)−F2(ν, σ)}.Proposition 3.3.1. Let F1 be a backward convex transfer on P(X1) ×P(Y1) with operators (F−1,i)i∈I , and let F2 be a backward convex couplingon P(X2)× P(Y1) with operators (F−2,j)j∈J . Theninfσ∈P(Y1){F1(µ, σ)−F2(ν, σ)} = infg∈C(X2){− infi∈I∫X1F−1,igdµ+ infj∈J∫X2F−2,jgdν}Proof. We have from the expression for F2 as a backward convex coupling,infσ∈P(Y1){F1(µ, σ)−F2(ν, σ)} = infσ∈P(Y1)infg∈C(X2)infj∈J{F1(µ, σ)−∫Y1gdσ +∫X2F−2,jgdν}= infg∈C(X2){− supσ∈P(Y1){∫Y1gdσ −F1(µ, σ)}+ infj∈J∫X2F−2,jgdν}= infg∈C(X2){− infi∈I∫X1F−1,igdµ+ infj∈J∫X2F−2,jgdν}where the last equality follows since F1 is a backward convex transfer.Corollary 3.3.2. Let F1 be a backward convex transfer on P(X1)× P(Y1)with operators (F−1,i)i∈I , and let F2 be a backward convex coupling on P(X2)×P(Y1) with operators (F−2,j)j∈J .Then the following are equivalent.781. For all σ ∈ P(Y1), we have F2(ν, σ) ≤ F1(µ, σ).2. For all g ∈ C(Y1), we have infi∈I∫X1F−1,i(g)dµ ≤ infj∈J∫X2F−2,j(g)dν.Proof. From Proposition 3.3.1, we have0 ≤ infσ∈P(Y1){F1(µ, σ)−F2(ν, σ)}if and only if0 ≤ infg∈C(X2){− infi∈I∫X1F−1,igdµ+ infj∈J∫X2F−2,jgdν}.This equivalence then reduces to the equivalence between 1. and 2.Corollary 3.3.3. Specializing Corollary 3.3.2 to the case where F1 = λE ?T(and relabeling F2 to F), for E a backward β-entropic transfer on P(X1)×P(Y1) with operator E−, T a backward linear transfer on P(Y1) × P(Y2)with operator T−, and λ > 0, then, for any fixed pair of probability measuresµ ∈ P(X1) and ν ∈ P(X2), the following are equivalent:1. For all σ ∈ P(Y2), we have F(ν, σ) ≤ λ E ? T (µ, σ).2. For all g ∈ C(X2), we have β( ∫X1E−◦T−g dµ) ≤ infi∈I1λ∫X2F−i (λg)dν.Proof. Note that the inf-convolution λE ? T is a backward (λβ)-entropictransfer (in particular, a backward convex transfer) with operator g 7→ E− ◦T−( gλ), therefore the above Corollary 3.3.2 applies and we obtain the dualinequalityλβ(∫X1E− ◦ T−( gλ)dµ)≤ infi∈I∫X2F−i (g)dν for all g ∈ C(X2).Dividing by λ and relabeling h := gλ , the inequality has to hold for allh ∈ C(X2), which completes the equivalence.We now apply the above to the case where E is the relative entropy H,in which case β(t) = log(t) and E−g = eg.79Corollary 3.3.4. In the setting of the above corollary, if we take E = H therelative entropy, then, for any fixed pair of probability measures µ ∈ P(X1)and ν ∈ P(X2), the following are equivalent:1. For all σ ∈ P(Y2), we have F(ν, σ) ≤ λH ? T (µ, σ)2. For all g ∈ C(X2), we have log( ∫X1eT−g dµ) ≤ infi∈I1λ∫X2F−i (λg)dν.Corollary 3.3.5. In the setting of Corollary 3.3.3, taking F = E2 to bea backward β2-entropic transfer on P(X2) × P(Y2) with operator E−2 , thefollowing are equivalent:1. For all σ ∈ P(Y2), we have E2(ν, σ) ≤ λ E1 ? T (µ, σ).2. For all g ∈ C(X2),β1( ∫X1E−1 ◦ T−g dµ) ≤ 1λβ2(∫X2E−2 (λg)dν).3.3.1 Moment measuresThe equivalences stated in Proposition 3.3.2 and following corollaries, wasthe writing of the duality for infσ∈P(Y1){F1(µ, σ) − F2(ν, σ)}. It turns outthat for a particular choice of F1 and F2, this duality is related to the studyof moment measures, which we briefly highlight here.Given a convex function u : Rd → R ∪ {+∞} with ∫Rd e−u(x)dx < +∞,its moment measure is defined as the measure µ := (∇u)#ρ, where dρ(x) =e−u(x)dx. Conversely, we say µ is a moment measure if there exists a convexfunction u such that µ = (∇u)#ρ. In [18], the authors provide sufficient andnecessary conditions for a probability measure µ to be the moment measureof some convex function.To prove the existence of a convex function u whose moment measureis the given measure µ, one can introduce, as was done by Santambrogio, avariational problem involving quadratic optimal transport and relative en-tropy. Indeed, the connection to quadratic optimal transport is via Brenier’stheorem, namely that µ = (∇u)#ρ implies ∇u is the optimal transport map80for the transport of ρ to µ with quadratic cost c(x, y) = |x − y|2. At thesame time, in light of the exponential form of the density dρ = e−u(x)dx,one is also not surprised that the logarithmic entropy also plays a role.To be precise, let H(dx, ν) := ∫Rd log(dνdx)dν if ν ∈ P(Rd) is absolutelycontinuous with respect to Lebesgue measure, and +∞ otherwise. Recallingfrom Example 2.6.7, the Brenier transferT (µ, ν) := infpi∈K(µ,ν)∫Rd×Rd−〈x, y〉dpi(x, y),consider the probleminfσ∈P1(Rd){H(dx, σ)− T (σ, µ)}.If we allow ourselves the non-compact setting of Rd, an immediate applica-tion of (4) in Proposition ?? is the following result in [18]infσ∈P(Rd){H(dx, σ)−W2(σ, µ¯)} = inff∈C(Rd){− log∫e−f∗dx+∫f dµ},where C(Rd) is the cone of convex functions on Rd, W2(σ, µ¯) is the Breniertransfer of Example 2.6.7, and µ¯ is defined as∫Rd f(x)dµ¯(x) =∫Rd f(−x)dµ(x).Note that in this case, T+f(x) = −f∗(−x), E−f = ef and β(t) = log t, andsince g∗∗ ≤ g,infσ∈P(Rd){H(dx, σ)−W2(σ, µ¯)} = H ? (−W2)(dx, µ¯)= inf{− log∫e−g dx+∫g∗(x) dµ; g ∈ C(Rd)}= inf{− log∫e−f∗dx+∫f dµ; f ∈ C(Rd)}.We remark that a particular interest is the characterization of those mea-sures µ (the moment measures) for which there is attainment in both min-imisation problems (see Cordero-Erausquin and Klartag [18]).813.4 Maurey-type inequalitiesProposition 3.4.1. Let F1 be a backward convex transfer on P(X1)×P(Y1)with operators (F−1,i)i∈I , F2 a backward convex transfer on P(X2) × P(Y2)with operators (F−2,j)j∈J , and F3 a backward convex coupling on P(Y1) ×P(Y2) with operators (F−3,k)k∈K . Then for fixed µ ∈ P(X1) and ν ∈ P(X2),the following holds:infσ1∈P(Y1),σ2∈P(Y2){F1(µ, σ1)−F3(σ1, σ2) + F2(ν, σ2)}= infg∈C(X2){− supk∈Kinfi∈I∫X1F−1,i(−F−3,k(g)) dµ− infj∈J∫X2F−2,j(g) dν}.Proof. Since as a convex coupling,F3(σ,σ2) = supg∈C(Y2){∫Y2gdσ2 − infk∈K∫Y1F−3,kgdσ1,we can substitute this expression into the following inf-convolution,infσ1∈P(Y1),σ2∈P(Y2){F1(µ, σ1)−F3(σ1, σ2) + F2(ν, σ2)}= infσ1∈P(Y1),σ2∈P(Y2)infg∈C(Y2){F1(µ, σ1)−∫Y2gdσ2 + infk∈K∫Y1F−3,kgdσ1 + F2(ν, σ2)}= infg∈C(Y2){− supk∈Ksupσ1∈P(Y1){∫Y1(−F−3,kg)dσ1 −F1(µ, σ1)} − supσ∈P(Y2){∫Y2gdσ2 −F2(ν, σ2)}}= infg∈C(Y2){− supk∈Kinfi∈I∫X1F−1,i(−F−3,kg)dν − infj∈J∫X2F−2,j(g)dν}Corollary 3.4.2. Let F1 be a backward convex transfer on P(X1)× P(Y1)with operators (F−1,i)i∈I , F2 a backward convex transfer on P(X2) × P(Y2)with operators (F−2,j)j∈J , and F3 a backward convex coupling on P(Y1) ×P(Y2) with operators (F−3,k)k∈K . Then for fixed µ ∈ P(X1) and ν ∈ P(X2),the following are equivalent:821. For all σ1 ∈ P(Y1), σ2 ∈ P(Y2), we haveF3(σ1, σ2) ≤ F1(µ, σ1) + F2(ν, σ2).2. For all g ∈ C(Y2)supk∈Kinfi∈I∫X1F−1,i(−F−3,k(g)) dµ+ infj∈J∫X2F−2,j(g) dν ≤ 0.Proof. By the previous Proposition 3.4.1, we have0 ≤ infσ1∈P(Y1),σ2∈P(Y2){F1(µ, σ1)−F3(σ1, σ2) + F2(ν, σ2)}if and only if0 ≤ infg∈C(X2){− supk∈Kinfi∈I∫X1F−1,i(−F−3,k(g)) dµ− infj∈J∫X2F−2,j(g) dν}.Corollary 3.4.3. Specializing Corollary 3.4.2 to the case where F1 = E1?T1and F2 = E2 ? T2 (and relabeling F3 to F), where• E1 is a backward β1-entropic transfer on P(X1)×P(Z1) with operatorE−1 ,• E2 is a backward β2-entropic transfer on P(X2)×P(Z2) with operatorE−2 ,• T1 is a backward linear transfer on P(Z1) × P(Y1) with operator T−1 ,and• T2 is a backward linear transfer on P(Z2)× P(Y2) with operator T−2 ,then, for any given λ1, λ2 ∈ R+ and (µ, ν) ∈ P(X1)× P(X2), the followingare equivalent:1. For all σ1 ∈ P(Y1), σ2 ∈ P(Y2), we haveF(σ1, σ2) ≤ λ1E1 ? T1(µ, σ1) + λ2E2 ? T2(ν, σ2).832. For all g ∈ C(Y2), we haveλ1 supk∈Kβ1(∫X1E−1 ◦ T−1 ◦ (−1λ1F−k g) dµ)+λ2β2(∫X2E−2 ◦ T−2 (1λ2g) dν)≤ 0.By applying the above to Ei = H the logarithmic entropy (i.e. whereβi(t) = log(t) and operator E−i f = ef ), we get the following extension of acelebrated result of Maurey [44].Corollary 3.4.4. Let E1 and E2 in the above Corollary 3.4.3 be equal to thelogarithmic entropy H. Then the following are equivalent:1. For all σ1 ∈ P(X1), σ2 ∈ P(X2), we haveF(σ1, σ2) ≤ λ1H ? T1(µ, σ1) + λ2H ? T2(ν, σ2).2. For all g ∈ C(Y2), we havesupk∈K(∫X1eT−1 (− 1λ1 F−k g) dµ)λ1 (∫X2eT−2 (1λ2g)dν)λ2≤ 1.If T1 = T2 are the identity transfer, then item 1 in the above is equivalent tosaying that for all g ∈ C(Y2), we havesupk∈K(∫X1e− 1λ1F−k g dµ)λ1 (∫X2e1λ2gdν)λ2≤ 1.84Chapter 4Hamiltonian/Lagrangiandynamics: Weak KAM,Aubry-Mather, and optimaltransport4.1 IntroductionIn this chapter, we provide an overview of weak KAM theory and Aubry-Mather theory for a Hamiltonian/Lagrangian system, drawing from [15, 19–21, 23, 32, 54, 55]. We shall see that the work of Bernard-Buffoni [5, 6]demonstrates that in fact much of the analysis can be obtained by studyingan optimal transport problem with the cost function of Example 2.6.6,cL(x, y) = inf{∫ 10L(γ(t), γ˙(t))dt ; γ ∈ C1([0, 1];M), γ(0) = x, γ(1) = y}.Inspired by the theory presented in this chapter, we shall then endeavourin the next chapter to develop a “weak KAM/Aubry-Mather” theory forgeneral linear transfers. Thus, the main purpose here is to describe theconnections with the aim of generalising them. For brevity, we shall not85attempt to provide complete proofs of results from the literature, and forpedagogical reasons, describe the results for the flat n-dimensional torusTn, although a smooth compact connected Riemannian manifold withoutboundary would suffice (with the appropriate changes).4.2 Hamiltonian systemsConsider a dynamical system on Tn := Rn/Zn, the n-dimensional flat torus,whose energy is described by a Hamiltonian function H : Tn × Rn → R,(x, p) 7→ H(x, p), where Rn is Euclidean space. The dynamics of the systemare obtained by solving Hamilton’s system of equations; a coupled, non-linear, system of ODE’s, given (at least in local coordinates) byx˙ = ∂H∂p (x, p)p˙ = −∂H∂x (x, p), (4.1)together with some initial conditions x(0) = x0 and p(0) = p0.Solving (4.1) explicitly is not, in general, possible, except in certainspecial cases. One such case, the case of an integrable system, occurs if,in the coordinates (x, p), the Hamiltonian H does not depend on x, i.e.H = H(p). Then the system reduces tox˙ = ∂H∂p (p)p˙ = 0, (4.2)which is easily solved and the dynamics is completely known: it correspondsto a uniform rotation on Tn given byx(t) = (x0 + ρ(p0)t) (mod Zn),with rotation vector ρ(p0) :=∂H∂p (p0). In particular, the phase space Tn×Rnis foliated by invariant tori Tn × {p0} which describe stable motions of thesystem, i.e. trajectories starting in an invariant tori will remain there for alltime. Different initial conditions of (4.2) will therefore trace out different86invariant tori in phase space. In addition, if the rotation vector ρ(p0) hasirrational coordinates (resp., rational), then the trajectory t 7→ x(t) on Tnis quasi-periodic (resp., periodic).It is then natural to ask about the dynamics of an integrable system ifit is slightly perturbed. What happens to the invariant tori and the stablemotions? Are they destroyed, or might some still persist?These questions are related to the search for a smooth and invertiblechange of coordinates Φ : (X,P ) 7→ (x, p) so that the Hamiltonian system(4.1) reduces to an integrable system in the new coordinates (X,P ):X˙ = ∂H¯∂P (P )P˙ = 0, (4.3)where the effective Hamiltonian H¯(P ) := H ◦ Φ(X,P ) depends only on P .In general, of course, such a coordinate change does not exist, for the fol-lowing reason: By comparing (4.1) and (4.3), a necessary condition relatingthe coordinates, is∂X∂x=∂p∂P.The above condition is satisfied if there exists a generating function u so thatX = ∂u∂p (x, P ) and p =∂u∂x(x, P ). The relation H¯(P ) = H(x, p) then yieldsan equation for this function u, the (stationary) Hamilton-Jacobi equation,H(x,∇xu(x, P )) = H¯(P ). (4.4)Thus the strategy becomes: Given P ∈ Rn, find a constant H¯(P ) and afunction u(·, P ) which satisfies (4.4), and define a change of coordinates(x, p)→ (X,P ) to satisfyX = ∇Pu(x, P )p = ∇xu(x, P ). (4.5)There are two main issues which prevent this procedure from being carried87out in general: The first is that the Hamilton-Jacobi equation (4.4) may notadmit any smooth solutions u, and second, even if it does, it may not bepossible to invert (4.5) globally to solve for P (and X) in terms of x and p.In the nearly integrable case, that is, whenH is “close” to some integrableH0, the theory contributed by Kolmogorov, Arnold, and Moser, collectivelyknown as KAM theory (see e.g. [58], [61], and references therein) providesresults which say that such a coordinate change does in fact exist. Someof the invariant tori of the unperturbed system are “deformed” and survivethe perturbation, while others are destroyed. The surviving tori are thoseof the unperturbed system which satisfy a “non-resonance condition”, madeprecise in the following (pseudo-) theorem.Theorem 4.2.1 (KAM theorem, [58], Theorem 2.1). Let H(x, p, ) :=H0(p) + H1(x, p, ) be a real-analytic Hamiltonian, where (x, p) ∈ Tn × Rnand  ∈ R. Suppose that the unperturbed system H0 has invertible Hes-sian at a point p0, where p0 is such that its corresponding rotation vec-tor ρ(p0) =∂H0∂p (p0) is diophantine, i.e. there exist c, γ > 0 such that|〈k, ρ(p0)〉| ≥ 1c|k|γ for all non-zero k ∈ Zn.Then for small  the torus Tn ×{p0} of the unperturbed system survivesand is slightly deformed by the perturbation. It carries trajectories withrotation vector ρ(p0).For more general H which are “far” from being integrable, one can-not hope to obtain similar results in this non-perturbative setting; in otherwords, the Hamiltonian dynamics are not conjugate to a rigid rotation andthere may not exist any quasi-periodic solutions. However, although theremay not exist classical C1 solutions to the stationary Hamilton-Jacobi (4.4),Lions, Papanicolaou, and Varadhan [39], have shown the existence of viscos-ity solutions to an equivalent formulation (sometimes called the cell prob-lem),H(x, P +∇xw(x, P )) = H¯(P ). (4.6)The equivalence of (4.6) and (4.4) is via u(x, P ) = x · P + w(x, P ).Theorem 4.2.2 ([39], Theorem 1). Suppose H is superlinear, i.e. lim|p|→∞H(x, p) =88+∞, uniformly in x. Then for each P ∈ Rn, there exists a unique H¯(P ) forwhich there exists a viscosity solution w of (4.6). In addition, P 7→ H¯(P )is continuous.It is often sufficient to consider in the next sections only the case P =0, the general case for any P obtained by replacing H with HP (x, p) :=H(x, P + p).Let us recall here the definition of a viscosity solution in this context.Definition 4.2.3 (Viscosity solution). Let k ∈ R.1. u ∈ C(Tn) is a viscosity subsolution of H(x,∇u(x)) = k iff forevery ϕ ∈ C1(Tn) such that ϕ ≥ u and every x0 ∈ Tn such thatϕ(x0) = u(x0), it holds thatH(x0,∇ϕ(x0)) ≤ k.2. u ∈ C(Tn) is a viscosity supersolution of H(x,∇u(x)) = k iff forevery ϕ ∈ C1(Tn) such that ϕ ≤ u and every x0 ∈ Tn such thatϕ(x0) = u(x0), it holds thatH(x0,∇ϕ(x0)) ≥ k.3. u ∈ C(Tn) is a viscosity solution iff u is both a visocity subsolution,and a viscosity supersolution.How is the effective Hamiltonian H¯(P ) related to the original system(4.1) when there exist only viscosity solutions? This is the weak KAM theoryof Fathi [20] that we discuss in the next section. It connects H¯(P ) and theassociated viscosity solutions, to the theory of invariant sets of Aubry-Matherfrom dynamical systems.Indeed, the lack of smooth solutions to Hamilton-Jacobi, implies non-existence of invariant tori. But we shall see in the next section that one canstill speak of more general sets which are invariant under the Hamiltonian89flow; this is Aubry-Mather theory. On the other hand, a result of Hamilton-Jacobi implies that solutions to the Hamilton-Jacobi equation give invariantsets.Theorem 4.2.4 (Hamilton-Jacobi). Let u : Tn → R be C2. Then for everyP ∈ Rn, Graph(P + ∇u) is invariant under the Hamiltonian flow, if andonly if H(x, P +∇u(x)) = c for a constant c independent of x.The basic premise of weak KAM theory is therefore to build the Aubry-Mather sets from viscosity solutions to the Hamilton-Jacobi equation.In the sections that follow the appropriate Hamiltonians are the TonelliHamiltonians; that is, those H : Tn × Rn → R satisfying1. (x, p) 7→ H(x, p) is Ck smooth for some k ≥ 2,2. (Strict Convexity) p 7→ H(x, p) is strictly convex for every fixed x,3. (Superlinearity) lim|p|→+∞H(x,p)|p| = +∞ uniformly in x.A Tonelli Hamiltonian defines, dually, a Tonelli Lagrangian L : Tn×Rn → Rvia the Fenchel-Legendre transform of H:L(x, v) := supp∈Rn{〈p, v〉 −H(x, p)}. (4.7)The Lagrangian L and Hamiltonian H are completely equivalent in the sensethat H can be recovered from L viaH(x, p) = supv∈Rn{〈p, v〉 − L(x, v)} ,and the statements “H is Tonelli”, and “L is Tonelli”, are equivalent, wherea Tonelli Lagrangian is a function L : Tn × Rn → R satisfying1. (x, v) 7→ L(x, v) is Ck smooth for some k ≥ 2,2. (Strict Convexity) v 7→ L(x, v) is strictly convex for every fixed x,3. (Superlinearity) lim|v|→+∞L(x,v)|v| = +∞ uniformly in x.90The transformation L(x, v) := (x, ∂L∂v (x, v)) is a global Ck−1 diffeomorphismbetween the Lagrangian coordinates (x, v) and the Hamiltonian coordinates(x, p), so thatH ◦ L(x, v) = 〈∂L∂v(x, v), v〉 − L(x, v),with inverse given by L−1(x, p) = (x, ∂H∂p (x, p)). The Hamiltonian system(4.1), yields in Lagrangian coordinates, the Euler-Lagrange equations,x˙ = v,ddt(∂L∂v (x, v))= ∂L∂x (x, v).(4.8)The Hamiltonian flow φHt associated to the Hamiltonian system (4.1) isconjugated to the Lagrangian flow φLt , via φHt = L ◦ φLt ◦ L−1.4.3 Aubry-Mather theory4.3.1 Man˜e´ critical valueDefinition 4.3.1. The Man˜e´ critical value, denoted by c, is the infimumof all k ∈ R such that there exists a continuous function u : Tn → R whichis a viscosity subsolution to H(x,∇u(x)) = k.Note that c ∈ R, since by superlinearity, there exists K > 0 and C > 0,such that L(x, v) ≥ K|v| − C, in which caseH(x, 0) = supv∈Rn{−L(x, v)} ≤ supv∈Rn{−K|v|+ C} = Chence u ≡ 0 is a viscosity subsolution for any k ≤ C. At the same timec > −∞ since for any u ∈ C1(Tn),H(x,∇u(x)) = supv∈Rn{〈∇u(x), v〉−L(x, v)} ≥ −L(x, 0) ≥ − supx∈TnL(x, 0) > −∞,thus there exists no viscosity subsolution of H(x,∇u(x)) = k, for k <− supx∈Tn L(x, 0). We shall see that in fact c = H¯(0), so the Man˜e´ crit-ical value is exactly the effective Hamiltonian (recall it suffices to consider91P = 0). The Legendre transform yields an important variational character-isation of viscosity subsolutions to the Hamilton-Jacobi equation which isrecorded in the following proposition.Proposition 4.3.2. u ∈ C(Tn) is a viscosity subsolution to H(x,∇u(x)) =c if and only if for any a < b ∈ R and every Lipschitz curve γ : [a, b]→ Tn,it holds thatu(γ(b))− u(γ(a)) ≤∫ baL(γ, γ˙)ds+ c(b− a). (4.9)Note in particular that this implies every viscosity subsolution is Lips-chitz by taking a curve γ with constant speed. The next proposition showsthat in fact if there is a γ for which there is equality in (4.9), then u isactually a viscosity solution and not just a subsolution.Proposition 4.3.3. If u is a viscosity subsolution to H(x,∇u(x)) = c, andfor every x ∈ Tn, there exists a Lipschitz curve γ : (−∞, 0] → Tn, withγ(0) = x, satisfyingu(γ(b))− u(γ(a)) =∫ baL(γ, γ˙)ds+ c(b− a)for all a < b ≤ 0, then u is a viscosity solution to H(x,∇u(x)) = c (andconsequently, c = H¯(0) is the unique constant for which this is true).The previous two propositions suggest the introduction of the Lax-Oleiniksemi-groupS−t u(x) := infγ {u(γ(−t)) +∫ 0−tL(γ, γ˙)ds}, t > 0, (4.10)where the infimum is over all Lipschitz curves γ such that γ(0) = x, andconsider those functions u for whichS−t u(x) + ct = u(x).92Definition 4.3.4. A function u ∈ C(Tn) is a negative weak KAM so-lution if S−t u(x) + ct = u(x) for all x ∈ Tn and all t > 0.The term “negative” refers to the fact that the Lipschitz curve in Propo-sition 4.3.3 is defined for all negative time. In view of Proposition 4.3.3,negative weak KAM solutions are in fact exactly the viscosity solutions forthe Hamilton-Jacobi equation.Proposition 4.3.5. u ∈ C(Tn) is a negative weak KAM solution if andonly if it is a viscosity solution to H(x,∇u(x)) = c.Remark 4.3.6. In a similar way to above, we may introduce another Lax-Oleinik semi-group (see (4.9) above),S+t u(x) := supγ{u(γ(t))−∫ t0L(γ, γ˙)ds}, t > 0,where the supremum is over all Lipschitz curves γ : [0, t] → R such thatγ(0) = x, and define the positive weak KAM solutions as those u satisfyingS+t u(x)− ct = u(x) for all x ∈ Tn and all t > 0.The positive weak KAM solutions are also viscosity solutions toH(x,∇u(x)) =c. We shall see the connection between positive/negative weak KAM solu-tions in the next sections.4.3.2 Peierls Barrier and the Aubry setIn this section, we will construct an invariant set under the Hamiltonianflow, the Aubry set, via the Peierls Barrier.Related to the Lax-Oleinik semi-group (see (4.10)),consider the minimal action,ht(x, y) := infγ{∫ t0L(γ, γ˙)ds ; γ(0) = x, γ(t) = y}(4.11)where the infimum is over all Lipschitz curves γ : [0, t] → Tn with thespecified end values. It is known that a minimiser of (4.11) exists, and93that it is as regular as L itself; this is Tonelli’s theorem. Moreover, by theCalculus of Variations, a minimiser of (4.11) satisfies the Euler-Lagrangeequation (4.8), which is nothing but the first variation of the action (4.11).Definition 4.3.7. The Peierls Barrier h∞(x, y) is defined viah∞(x, y) = lim inft→+∞ (ht(x, y) + ct),where c = H¯(0) is Man˜e´’s constant of the previous section.Proposition 4.3.8. The Peierls Barrier is finite for all x, y ∈ Tn, Lipschitz,and satisfies the following properties:1. Every viscosity subsolution u satisfies u(x)− u(y) ≤ h∞(x, y),2. h∞(x, z) ≤ h∞(x, y) + h∞(y, z),3. h∞(x, y) + h∞(y, x) ≥ 0.The Peierls Barrier is important for a number of reasons, the first beingthat it provides negative weak KAM solutions.Proposition 4.3.9. For every fixed x ∈ Tn, the map y 7→ h∞(x, y) is anegative weak KAM solution.The collection of points where the Peierls Barrier vanishes is particularlyimportant as we will see next.Definition 4.3.10. The projected Aubry set is defined as the collectionof points where the Peierls barrier vanishes:A := {x ∈ Tn ; h∞(x, x) = 0}.Proposition 4.3.11. The projected Aubry set A is compact and non-empty.Proposition 4.3.12. The projected Aubry set satisfiesA =⋂u(u− S−1 u)−1(c)where the intersection is over all viscosity subsolutions u to H(x,∇u(x)) = c.94The projected Aubry set is distinguished for a number of reasons. Thefirst we shall note is that it a set of “uniqueness” for weak KAM solutions.Proposition 4.3.13. 1. If u is a negative weak KAM solution, thenthere exists a unique positive weak KAM solution v such that u = von A.2. If u is a viscosity subsolution of H(x,∇u(x)) = c, then there exists anegative weak KAM solution v such that u = v on A.3. If u1, u2 are negative weak KAM solutions, and u1 ≤ u2 on A, thenu1 ≤ u2 everywhere.The second we shall observe is that it is exactly the differentiability setfor viscosity subsolutions of the Hamilton-Jacobi equation.Proposition 4.3.14. For any x ∈ A, there exists a C2 curve γx : R → Awith γx(0) = x, that solves the Euler-Lagrange equation,ddt(∂L∂v(γx, γ˙x))=∂L∂x(γx, γ˙x), for all t ∈ Rfor which any viscosity subsolution u satisfiesu(γx(b))− u(γx(a)) =∫ baL(γx, γ˙x)ds+ c(b− a), a < b ∈ R. (4.12)In addition, u is differentiable at x with ∇u(x) = ∂L∂v (x, γ˙x(0)) and satisfiesH(x,∇u(x)) = c at that point.The curve γ is unique in the sense that if γ˜x : [a, b] → Tn is defined onan interval [a, b] containing 0, satisfies γ˜x(0) = x and (4.12), then γx = γ˜xon [a, b].The above proposition implies that the map x 7→ ∂L∂v (x, γ˙x(0)) is well-defined on A. We define the graph of this function over A to be the Aubryset.95Definition 4.3.15. The Aubry set A˜ is the subset of Tn×Rn defined viaA˜ := {(x, ∂L∂v(x, γ˙x(0))) ; x ∈ A}where γ is as in Proposition 4.3.14.Proposition 4.3.16. The Aubry set is non-empty, compact, and invariantunder the Hamiltonian flow. It is a Lipschitz graph over the projected Aubryset.4.3.3 Mather measuresRecall the minimal action,ht(x, y) := infγ{∫ t0L(γ, γ˙)ds}where the infimum is over Lipschitz curves γ : [0, t]→ Tn with γ(0) = x andγ(t) = y.In the previous section, the Peierls Barrier was defined as a long-timelimit of the minimal action, and the Aubry set was constructed. However,one alternative to studying minimising trajectories of (4.11), is instead lookat the average action of a “collection of orbits”, in the sense of a Lagrangianaction of a flow-invariant probability distribution:AL[µ] :=∫Tn×RnL(x, v)dµ(x, v),and introduce the analogous minimisation problem,inf {AL[µ] ; µ ∈ D} ,where D := {µ ∈ P(Tn × Rn) ; (φLt )#µ = µ} is the set of probability mea-sures invariant under the Euler-Lagrange flow determined by L.Since the constraint set D depends on L itself, Man˜e´ introduced a largerset of probability measures, the so-called set of holonomic measures F , which96contains D, and is defined byF :={µ ∈ P(Tn × Rn) ;∫Tn×Rnv · ∇ϕ(x)dµ(x, v) = 0, ∀ϕ ∈ C1(Tn)}.The corresponding minimisation problem,inf {AL[µ] ; µ ∈ F} (4.13)is a relaxation of (4.11).How do minimisers of (4.13) relate to trajectories of (4.1)? Consider theintegrable Hamiltonian system (4.2). In this case, L(x, v) = L(v), and byFenchel-Legendre (4.7), we have for any p0 ∈ Rn,L(v) ≥ 〈p0, v〉 −H(p0)with equality when v = ∂H∂p (p0) = ρ(p0). Integrating with respect to µ ∈ F ,we obtain∫Tn×RnL(v)dµ(x, v) ≥ 〈p0,∫Tn×Rnvdµ(x, v)〉 −H(p0) (4.14)which can be rearranged to yield the inequality∫Tn×Rn[L(v)− 〈p0, v〉]dµ(x, v) ≥ −H(p0). (4.15)This linear shift of the Lagrangian, Lp0(v) := L(v)−〈p0, v〉, is again a TonelliLagrangian with the same Euler-Lagrange equation as L.We recall the invariant tori Tn×{p0} for the integrable Hamiltonian sys-tem. These translate via the Fenchel-Legendre transform (4.7) into invarianttori Tn × {ρ(p0)} for the corresponding Euler-Lagrange flow. If µ ∈ F isnow supported on Tn × {ρ(p0)}, then the left-hand side of (4.15) becomes∫Tn×Rn[L(ρ(p0))− 〈p0, ρ(p0)〉]dµ(x, v) = −H(p0).The conclusion is the following: Any invariant measure µ ∈ F supported on97Tn ×{ρ(p0)} minimises the Lagrangian Lp0 among all measures µ ∈ F , i.e.,inf{ALp0 [µ] ; µ ∈ F} = −H(p0), (4.16)and, in addition,Tn × {ρ(p0)} =⋃{µ∈F ;µ minimises ALp0 }spt(µ) (4.17)The action-minimising measures µ thus provide a characterisation for theinvariant tori in the case of an integrable system.Still considering for the moment the case of an integrable system, definean average rotation vector for µ, by ρ(µ) :=∫Tn×Rn vdµ(x, v) ∈ Rn, whichcan be interpreted as a kind of average over the rotation vectors of a collec-tion of orbits. From (4.14), we have for measures µ ∈ F with ρ(µ) = ρ(p0),∫Tn×RnL(v)dµ(x, v) ≥ 〈p0, ρ(p0)〉 −H(p0)= L(ρ(p0)),with equality if µ is supported in Tn × {ρ(p0)}. Therefore,inf{AL[µ] ; µ ∈ F , ρ(µ) = ρ(p0)} = L(ρ(p0)), (4.18)and moreover,Tn × {ρ(p0)} =⋃{µ∈F ;µ minimises AL and ρ(µ)=ρ(p0)}spt(µ). (4.19)Motivated by the integrable case (4.16), for general Hamiltonian systemsthe so-called Mather’s α-function is then defined viaα(c) := − inf{ALc [µ] ; µ ∈ F}, c ∈ Rn, (4.20)where we recall that Lc(x, v) := L(x, v)−〈v, c〉. In the case of an integrablesystem, Mather’s α-function is exactly the Hamiltonian (4.16); hence α is98also known as the effective Hamiltonian in the general case. In fact, we shallsee in Section 4.4 that a result of weak KAM says in fact that α(P ) = H¯(P )where we recall H¯(P ) is the effective Hamiltonian of Lions, Papanicolaou,and Varadhan.Let Mc denote the measures µ ∈ F whose action ALc [µ] realise theminimal value −α(c). As motivated by (4.17), define the Mather set M˜ccorresponding to c as the union of the support of all measures in Mc:M˜c :=⋃µ∈Mcspt (µ) ⊂ Tn × Rn (4.21)and the projected Mather set Mc as the projection of M˜c onto the base Tn:Mc := pi(M˜c) ⊂ Tn.Theorem 4.3.17 (Mather). M˜c is a compact, non-empty, and invariant setunder the Euler-Lagrange flow. Moreover, it is a graph over Mc.Analogous to Mather’s α-function, and again motivated by the integrablecase (4.18), Mather’s β-function is defined via,β(h) := inf{AL[µ] ; µ ∈ F , ρ(µ) = h}, h ∈ Rn, (4.22)where ρ(µ) is the rotation vector of µ. Since β coincides with the Lagrangianwhen the system is integrable, it is also termed the effective Lagrangian. Theterminology is suggestive that β should be the Fenchel-Legendre transformof the effective Hamiltonian, i.e., that β = α∗ (see Theorem 4.3.18 below).Similarly to Mather’s α-function, let Mh denote the measures µ ∈ Fwhose rotation vector ρ(µ) is equal to h, and whose action AL[µ] realisesthe minimum β(h) in (4.22). Define the Mather set M˜h as the union of thesupport of all measures in Mh:M˜h :=⋃µ∈Mhspt (µ) ⊂ Tn × Rn,99and its projection Mh as the projection of M˜h onto the base Tn:Mh := pi(M˜h) ⊂ Tn.Mather again proved that M˜h is a compact, non-empty, and invariant setunder the Euler-Lagrange flow. Moreover, it is a graph over Mh.Theorem 4.3.18. Mather’s functions α and β are convex conjugates ofeach other, i.e. α = β∗ and β∗ = α.In addition, if µ ∈ F is an invariant probability measure, thenβ(ρ(µ)) = AL[µ] ⇐⇒ there exists c ∈ Rn such that −α(c) = ALc [µ].Furthermore, if β(ρ(µ)) = AL[µ], then−α(c) = ALc [µ] ⇐⇒ c ∈ ∂β(ρ(µ)).These results imply that the two collections of minimising measures arethe same, ⋃c∈RnMc =⋃h∈RnMh,and moreover, the Mather sets are related viaM˜c =⋃h∈∂α(c)M˜h.Proposition 4.3.19. The Mather set is contained in the Aubry set, i.e.,M˜0 ⊂ A˜, where M˜0 is given by (4.21).Proposition 4.3.20. A probability measure µ belongs to M0 if and only ifits support is contained in the Aubry set A˜.4.4 Weak KAM theoryAs mentioned in the introduction, the goal of weak KAM theory is to connectviscosity solutions of the stationary Hamilton-Jacobi equation to Aubry-Mather; specifically the Mather measures, Mather set, and Aubry set.100The Aubry set was defined above via the Peierls Barrier. Here we showthat the Aubry set can in fact be constructed from the negative weak KAMsolutions.First, we have the following connection between the various functions.Proposition 4.4.1. Mather’s α-function, the effective Hamiltonian H¯ ofLions, Papanicolaou, and Varadhan, and Man˜e´’s critical constant c, are thesame, i.e. α(0) = H¯(0) = c.Proposition 4.4.2. For every negative weak KAM solution u, the setGraph(∇u) := {(x,∇u(x)) ; ∇u(x) exists },under backwards Hamiltonian flow satisfies,φH−t(Graph(∇u)) ⊂ Graph(∇u) for all t ≥ 0,and thereforeI˜(u) :=⋂t≥0φH−t(Graph(∇u))is invariant under the Hamiltonian flow. It is non-empty and compact, andknown as the Aubry set associated to u.Proposition 4.4.3. The Aubry set A˜ as defined in Section 4.3.2 is equalto the intersection over all I˜(u) for all negative weak KAM solutions u,A˜ =⋂uI˜(u).4.5 Connections of weak KAM andAubry-Mather to optimal transportRecall the minimal action ht(x, y) between two points x and y defined inSection 4.3.2,ht(x, y) := infγ{∫ t0L(γ, γ˙)ds ; γ(0) = x, γ(t) = y}(4.23)101where the infimum is over Lipschitz curves γ : [0, t] → Tn. Bernard andBuffoni [5, 6] noted that a lot of the analysis in the previous sections doesnot really depend on the explicit form of ht(x, y).To emphasise this, they develop an abstract theory based on a continuousfunction A : X ×X → R on a compact connected metric space X, of whichht for t = 1 in (4.23) is an example. They build an analogue of the PeierlsBarrier h∞ of the previous sections via inf-convolution of A with itself:An(x, y) = A ? A ? . . . ? A(x, y)= inf{A(x, x1) +A(x2, x3) + . . .+A(xn−1, y) ; xi ∈ X}.The cost An(x, y) between two points x and y, does not deviate too far fromthe long time average costc := limn→∞inf{An(x, y) ; x, y ∈ X}n= limn→∞sup{An(x, y) ; x, y ∈ X}n(4.24)in a uniform way, in the sense that |An(x, y) − nc| ≤ C for a constant Cindependent of x,y, and n.Lemma 4.5.1 ([6], Lemma 9). The functions An are equi-continuous, andthere exists a constant C > 0, such that|An(x, y)− nc| ≤ C for all x, y ∈ X, and n ∈ N.Similar to Section 4.3.2, one can define a notion of the Peierls Barrier inthis setting.Definition 4.5.2. The Peierls Barrier is the functionA∞(x, y) := lim infn→∞ (An(x, y)− nc).The function A∞ is continuous and real-valued as the family (An)n∈Nis equi-continuous and An − cn is uniformly bounded in x, y, and n. ThePeierls Barrier A∞ has the same properties as h∞ (cf. Proposition 4.3.8),102which are recorded in the following.Proposition 4.5.3 ([6], Lemma 11). The function A∞ satisfiesA∞(x, z) ≤ A∞(x, y) +A∞(y, z) for all x, y, z ∈ X,andA∞(x, z) = infy∈A{A∞(x, y) +A∞(y, z)}where A := {x ; A∞(x, x) = 0} is non-empty.Definition 4.5.4. The projected Aubry set is the setA = {x ; A∞(x, x) =0}.Recall from Proposition 4.3.8 that for any viscosity subsolution u, itheld that u(y) − u(x) ≤ h∞(x, y), i.e., u is h∞-Lipschitz. In the abstractsetting for A∞, functions u which are A∞-Lipschitz are exactly the admissi-ble functions for the Kantorovich dual problem corresponding to an optimaltransport with cost A∞ (see Section 2.6).Definition 4.5.5. Let φ0, φ1 ∈ C(X). We say (φ0, φ1) is a conjugate pair(for A∞) ifφ1(y) = T−A∞φ0(y) and φ0(x) = T+A∞φ1(x)whereT−A∞φ0(x) := supy∈X{φ0(y)−A∞(x, y)} and T+A∞φ1(y) := infx∈X{φ1(x)+A∞(x, y)}.We have the following proposition.Proposition 4.5.6 ([6], Proposition 8). If (φ0, φ1) are a conjugate pair forA∞, then there exists a A∞-Lipschitz function φ : A → R such thatφ0 = φ1 = φ on A,103and φ0, φ1 can recovered from φ viaφ1(y) = infx∈A{φ(x) +A∞(x, y)} and φ0(x) = supy∈A{φ(y)−A∞(x, y)}.(4.25)Conversely, for every A∞-Lipschitz function ϕ on A, the functions φ0, φ1defined by (4.25) are a conjugate pair for A∞.Remark 4.5.7. The above proposition is stated for A∞, but in fact it istrue for any c ∈ C(X×X) which satisfies the properties listed in Proposition4.5.3.As was the case in the previous sections, the projected Aubry set isparticularly important.Theorem 4.5.8 ([6], Theorem 12). (φ0, φ1) are a conjugate pair for A∞ ifand only if, T−A φ0 + c = φ0, T+A φ1 − c = φ1, and φ0 = φ1 on A, whereT−A φ0(x) := supy∈X{φ0(y)−A(x, y)} and T+A φ1(y) := infx∈X{φ1(x) +A(x, y)}and c is defined as in (4.24).Remark 4.5.9. This theorem is the abstract analogue of Proposition 4.3.13,item 1.Recall from Section 2.6 that conjugate pairs are connected to an optimaltransport viainfpi∈K(µ,ν)∫X×YA∞(x, y) dpi(x, y) = sup{∫Yφ1dν −∫Xφ0dµ}where the supremum is over (φ0, φ1) which are are conjugate pairs for A∞.Therefore the interpretation of this theorem is that taking the Peierls BarrierA∞ as the cost function for an optimal transport, the admissible pairs for theKantorovich duality are exactly the negative/positive weak KAM solutionswhich agree on the projected Aubry set. Indeed, those functions which104satisfyT−A φ0 + c = φ0 and T+A φ1 − c = φ1are the analogues of Fathi’s positive (resp., negative) weak KAM solutions inthis abstract setting (note here that one should think of T−A as correspondingto S+1 , while T+A as corresponding to S−1 , where S±1 are the Lax-Oleinikoperators of the previous section; the apparent discrepancy in +, −, notationis chosen to be consistent with backward/forward Kantorovich operators forthe next chapter) since the above relations implyT−A,nφ0 + nc = φ0 and T+A,nφ1 − nc = φ1, for all n ∈ Nwhere T±A,n denotes the n-fold composition of T±A . The main distinction isthat the “time” index here is integer-valued (compare with Definition 4.3.4).Finally the Mather measures are interpreted in this setting.Theorem 4.5.10 ([6], Theorem 13). Denote the optimal transport with costfunction A byTA(µ, ν) := infpi∈K(µ,ν)∫X×XA(x, y)dpi(x, y).Thenc = infµ∈P(X)TA(µ, µ),and a measure pi ∈ P(X ×X) satisfies ∫X×X A(x, y)dpi(x, y) = TA(µ, µ) = cif and only if pi is supported on the setD := {(x, y) ; A(x, y) +A∞(y, x) = c}.Remark 4.5.11. In the setting of the previous sections when X = Tn,there is a bijection between the Mather measures M0 and the measures pi,which is given by the mapping (projTn , projTn ◦ ϕL1 )#, where projTn is theprojection Tn × Rn → Tn, and ϕL1 is the time 1 Lagrangian flow.105Chapter 5Weak KAM andAubry-Mather for lineartransfersIn this chapter, our aim is the development of a “weak-KAM/Aubry-Mather”theory for a linear transfer in an analogy to the previous chapter. In par-ticular, for a backward linear transfer T : P(X)×P(X)→ R ∪ {+∞} withbackward Kantorovich operator T− : C(X) → USC(X), it is natural inview of the last chapter to define backward weak KAM solutions as thosefunctions g ∈ USC(X) satisfying T−g(x)+c = g(x) for a particular constantc (essentially the fixed points for T− up to an additive constant). An imme-diate technical issue is that T−g is not strictly defined when g ∈ USC(X);we therefore discuss its extension from C(X) in the next section, Section5.1. The backward weak KAM solutions can be viewed as a generalisa-tion of Fathi’s weak KAM solutions from the previous chapter, and indeedreduce to them when the linear transfer T is optimal transport generatedfrom a Lagrangian. If T is also a forward linear transfer with forwardKantorovich operator T+, one can speak of forward weak KAM solutionsT+f(x) − c = f(x), but since a backward linear transfer is not, in general,also a forward linear transfer, we shall focus mainly on backward weak KAMsolutions.106We shall see that for many backward linear transfers, we can constructan idempotent backward Kantorovich operator T−∞, which maps C(X) intothe set of backward weak KAM solutions for T−. We can also associate toT−∞ an idempotent backward linear transfer T∞, which in the case of optimaltransport generated from a Lagrangian, is related to the Peierls barrier fromthe previous chapter.5.1 Extension of Kantorovich operators fromC(Y ) to USCσ(Y )We recall that USC(X) is the set of all extended real-valued upper semi-continuous functions f : X → R ∪ {−∞}, and USCσ(X) the closure ofUSC(X) with respect to monotone increasing limits (i.e. f ∈ USCσ(X)if and only if there exists a monotone increasing sequence fn ∈ USC(X)with f(x) = limn→∞ fn(x) for all x ∈ X). We also denote USCb(X) asthose USC(X) functions which are bounded below (and therefore boundedabove and below by compactness of X). We shall also denote by USCσ,b(X)those functions in USCσ(X) which are bounded above and below. The sameconventions also hold for the space Y in place of X.As mentioned in the introduction of this chapter, we wish to find back-ward weak KAM solutions that are achieved by iterating T− with itself ntimes in the limit as n → ∞. Thus it is necessary to extend the domain ofthe Kantorovich operator T−, which as defined is only on C(X), to theselarger classes of functions. We note that the extension of T− to USCb(X)coincides with the discussion of Section 4.2 in the independent work [1].Lemma 5.1.1. Let T be a backward linear transfer such that {δx;x ∈ X} ⊂D1(T ), and let T− : C(Y ) → USC(X) be the associated backward Kan-torovich operator.1. For g ∈ USC(Y ), defineT−g(x) := inf{T−h(x) ; h ∈ C(Y ), h ≥ g}.107Then T− maps USC(Y ) into USC(X), and has representationT−g(x) = sup{∫Ygdν − T (δx, ν); ν ∈ P(Y )}. (5.1)Moreover, T− maps USCb(Y ) into USCb(X) if and only ifsupx∈Xinfν∈P(Y )T (δx, ν) < +∞.2. For g ∈ USCσ(Y ), defineT−g(x) := sup{T−h(x) ; h ∈ USC(Y ) , h ≤ g},where T−h is defined as in 1. Then T− maps USCσ(Y ) to USCσ(X).If, in addition, g ∈ USCσ(Y ) is bounded above, then (5.1) also holds.T− therefore extends to an operator from USCσ(Y ) → USCσ(X) satisfy-ing the monotonicity, convexity, and affine constant properties of Definition2.4.1, only on USCσ(Y ) rather than C(Y ).Proof. 1. The fact that T−g belongs to USC(X) follows from the fact thatan infimum of a family of upper semi-continuous functions, is also uppersemi-continuous.Suppose now that g ∈ USC(Y ). Let hn ↘ g be a decreasing sequenceof continuous functions converging to g. Then,T−g(x) ≤ T−hn(x) = supσ∈P(Y ){∫Yhndσ − T (δx, σ)} =∫Yhndσn − T (δx, σn),where the supremum is achieved for some probability measure σn becauseσ 7→ ∫Y hndσ−T (δx, σ) is weak∗ upper semi-continuous and bounded aboveon the compact set P(Y ).By weak∗ compactness of P(Y ), extract an increasing subsequence nkso that σnk → σ. Then for any j ≤ k, T−g(x) ≤∫Y hnjdσnk − T (δx, σnk)where we have used the fact that hnk ≤ hnj whenever j ≤ k. For this fixedj, we have that hnj ∈ C(Y ) and so∫hnjdσnk →∫hnjdσ as k →∞. Hence108we obtainT−g(x) ≤ limk→∞∫Yhnjdσnk − lim infk→∞T (δx, σnk) ≤∫Yhnjdσ − T (δx, σ).Finally∫Y hnjdσ →∫Y gdσ by monotone convergence, and we obtainT−g(x) ≤ supσ∈P(Y ){∫Ygdσ − T (δx, σ)}.On the other hand, for any h ∈ C(Y ), h ≥ g,supσ∈P(Y ){∫Ygdσ − T (δx, σ)} ≤ supσ∈P(Y ){∫Yhdσ − T (δx, σ)} = T−h(x).Therefore we obtain the reverse inequalitysupσ∈P(Y ){∫Ygdσ − T (δx, σ)} ≤ inf{T−h(x) ; h ∈ C(Y ), h ≥ g} = T−g(x).Suppose now that T−g ∈ USCb(X). Then using (5.1), we have−∞ < infx∈XT−g(x) = infx∈Xsupν∈P(Y ){∫Ygdν − T (δx, ν)}≤ supy∈Yg(y)− supx∈Xinfν∈P(Y )T (δx, ν)which implies supx∈X infν∈P(Y ) T (δx, ν) < +∞.On the other hand, if supx∈X infν∈P(Y ) T (δx, ν) < +∞, then we easilysee using (5.1) again thatinfx∈XT−g(x) ≥ infy∈Yg(y)− supx∈Xinfν∈P(Y )T (δx, ν) > −∞from which we deduce that T−g ∈ USCb(X).Note that T−g is bounded above since T−g(x) ≤ supy∈Y g(y) − mT ,where mT is a lower bound for T .2. For g ∈ USCσ(Y ), we have T−g ∈ USCσ(X) as it is by definition thesupremum of a family of upper semi-continuous functions. Now suppose g109is, in addition, bounded above. Then the expression∫Ygdσ − T (δx, σ)is well-defined for all σ ∈ P(Y ) and takes values in R∪{−∞}. Use now thefirst part to write for any h ∈ USC(Y ), h ≤ g,supσ∈P(Y ){∫Ygdσ − T (δx, σ)} ≥ supσ∈P(Y ){∫Yhdσ − T (δx, σ)} = T−h(x)which implies by definition of T−g thatsupσ∈P(Y ){∫Ygdσ − T (δx, σ)} ≥ T−g(x). (5.2)On the other hand, for an increasing hn ↗ g, hn ∈ USC(Y ),T−g(x) ≥ T−hn(x) ≥∫Yhndσ − T (δx, σ), for any σ ∈ P(Y ).By the monotone convergence of hn to g, we may take the limit as n→∞in the above inequality, and concludeT−g(x) ≥∫gdσ − T (δx, σ) for any σ ∈ P(Y ),whereby taking the supremum in σ yields the reverse inequality of (5.2) andgives the desired equality 5.1.Remark 5.1.2. Note that even though for g ∈ USCσ(Y ) which is boundedabove, we have the expression T−g(x) = supσ∈P(Y ){∫Y gdσ − T (δx, σ)},it does not necessarily imply that T−g ∈ USC(X). Indeed, the mapσ 7→ ∫Y gdσ has no weak∗ semi-continuity property in general when g ∈USCσ(Y ), so the supremum above may not be achieved.Also note that it may happen that both terms in the expression∫Y gdσ−T (δx, σ) are simultaneously +∞ for a given σ when g ∈ USCσ(Y ).110Lemma 5.1.3 (Monotone limits). Let T be a backward linear transfer asin Lemma 5.1.1, and let T− denote its corresponding Kantorovich operator,extended to USCσ(Y ).1. If gn, g ∈ USC(Y ) and gn ↘ g, then T−gn ↘ T−g.2. If gn, g ∈ USCσ(Y ) are bounded above with gn ↗ g, then T−gn ↗T−g.Proof. 1) We have by the monotonicity property of T−, that T−g(x) ≤lim infn T−gn(x). If for some n, T−gn(x) = −∞, then T−g(x) = −∞ andthere is nothing to prove. Otherwise, T−gn(x) > −∞ for all n, in whichcase the expression (5.1) is finite. The mapσ 7→∫Ygndσ − T (δx, σ)is weak∗ upper semi-continuous (since σ 7→ ∫Y hdσ is weak∗ upper semi-continuous for any h ∈ USC(Y )) , so it achieves its supremum at some σn,i.e.,T−gn(x) =∫gndσn − T (δx, σn).Extract an increasing subsequence nk so that lim supn T−gn(x) = limk T−gnk(x)and σnk → σ¯. Similarly to the proof of Lemma 5.1.1, by monotonicity ofgn, we haveT−gnk(x) ≤∫gnjdσnk − T (δx, σnk) for fixed j ≤ k. (5.3)As gnj ∈ USC(Y ) and σnk → σ¯, it follows that lim supk→∞∫gnjdσnk ≤∫gnjdσ¯. Hence upon taking lim supk→∞ in (5.3), we concludelim supnT−gn(x) ≤∫gnjdσ¯ − T (δx, σ¯).Then we let j →∞ and use monotone convergence to conclude thatlim supnT−gn(x) ≤ supσ∈P(Y ){∫gdσ − T (δx, σ)} = T−g(x).1112) Again, by the monotoncity property for T−, T−g ≥ lim supn T−gn(x).On the other hand, we know by Lemma 5.1.1 thatT−gn(x) = supσ∈P(Y ){∫Ygndσ − T (δx, σ)} ≥∫gndσ − T (δx, σ) for all σ.Hence by monotone convergence, lim infn T−gn(x) ≥∫gdσ−T (δx, σ) for allσ. Taking the supremum over σ yields lim infn T−gn(x) ≥ supσ∈P(Y ){∫Y gdσ−T (δx, σ)} = T−g(x).Remark 5.1.4. We note that the given proof of item (1) of the abovelemma fails if we allow sequences gn ∈ USCσ(Y ). This is because there isno weak∗ semi-continuity property for ν 7→ ∫Y gdν when g merely belongsto USCσ(Y ) and not USC(Y ).Corollary 5.1.5. Let T : P(X)×P(Y )→ R ∪ {+∞} be a backward lineartransfer such that {δx ; x ∈ X} ⊂ D1(T ). Then,1. For any (µ, ν) ∈ P(X)× P(Y ), we haveT (µ, ν) = sup{∫Yg dν −∫XT−g dµ; g ∈ USCb(Y )}= sup{∫Yg dν −∫XT−g dµ; g ∈ USCσ,b(Y )}.2. The Legendre transform formula (2.4) for Tµ, which holds for C(Y ),extends to USC(Y ); that is, for µ ∈ P(X) and all g ∈ USC(Y ), wedefineT ∗µ (g) := sup{∫Ygdσ − T (µ, σ);σ ∈ P(Y )}and then we haveT ∗µ (g) =∫XT−g dµ. (5.4)Proof. 1. For g ∈ USCb(Y ), take a monotone decreasing sequence gn ∈C(Y ) with gn → g. By Lemma 5.1.3, and monotone convergence, we inferthatlimn→∞(∫Ygndν −∫XT−gndµ)=∫Ygdν −∫XT−gdµ,112from which we concludeT (µ, ν) ≥ sup{∫Yg dν −∫XT−g dµ; g ∈ USCb(Y )}.The reverse inequality is immediate because C(Y ) ⊂ USCb(Y ).2. Let g ∈ USC(Y ) and take gn ↘ g with gn ∈ C(Y ). We have∫Ygndσ − T (µ, σ) ≤ sup{∫Ygndσ − T (µ, σ);σ ∈ P(Y )} =∫XT−gn dµso that by Lemma 5.1.3 upon taking n→∞, we obtain∫Ygdσ − T (µ, σ) ≤∫XT−g dµand consequentlysup{∫Ygdσ − T (µ, σ);σ ∈ P(Y )} ≤∫XT−g dµ. (5.5)On the other hand, by monotonicity, T−g ≤ T−gn, so∫XT−g dµ ≤∫XT−gn dµ ≤ sup{∫Ygndσ − T (µ, σ);σ ∈ P(Y )}.Since µ ∈ D1(T ), the supremum on the right-hand side is achieved by someσn. Extract an increasing subsequence nj so that σnj → σ for some σ ∈P(Y ). Then if i ≤ j, we have gnj ≤ gni , so that∫XT−g dµ ≤∫Ygnidσnj − T (µ, σnj ) for i ≤ jwhere upon sending j →∞ yields∫XT−g dµ ≤∫Ygnidσ − T (µ, σ)and finally i→∞ by monotone convergence yields the reverse inequality of(5.5).113Remark 5.1.6. For similar reasons as noted in a previous remark, the proofof item 1 in the above corollary fails if USCb(Y ) is replaced with USC(Y ),(similarly item 2 to USCσ(Y )), since it may happen that both∫Y gdν = −∞and∫X T−gdµ = −∞ simultaneously.The following is the extension of the inf-convolution stability of Corollary2.10.4 as mentioned in Remark 2.10.5.Corollary 5.1.7. Let T be a backward linear transfer on P(X)×P(X) with{δx ; x ∈ X} ⊂ D1(T ) and D1(T ) ∩D2(T ) 6= ∅. Then T ? T is a backwardlinear transfer with backward Kantorovich operator T− ◦ T−.Proof. The key point here is that T−g ∈ USC(X), so by Corollary 5.1.5 (inparticular, (5.4); replace g there by T−g for g ∈ C(X)), we conclude thatwhen µ ∈ D1(T ? T ),(T ? T )∗µ(g) = supν∈P(X){∫Xgdν − T ? T (µ, ν)}= supσ∈P(X){∫XT−gdσ − T (µ, σ)}=∫XT− ◦ T−gdµ, for all g ∈ C(X).5.1.1 Conjugate functions for forward and backward lineartransfersWe have the following notion motivated by the theory of mass transport (seeSection 2.6 and also 4.5).Definition 5.1.8. Let T be both a backward and forward transfer withKantorovich operators T− and T+. A pair (f, g) ∈ USCσ(X) × USCσ(Y )are a conjugate pair ifT−g = f and T+f = g.114The following proposition shows in particular that for any function g ∈C(Y ), the couple (T−g, T+ ◦ T−g) form a conjugate pair.Proposition 5.1.9. Suppose T : P(X) × P(Y ) → R ∪ {+∞} is both aforward and backward linear transfer, and that {(δx, δy) ; (x, y) ∈ X × Y } ⊂D(T ) such that T+ : C(X)→ C(Y ) and T− : C(Y )→ C(X). Then for anyg ∈ C(Y ) , (resp.,f ∈ C(X))T+ ◦ T−g ≥ g T− ◦ T+f ≤ f ,andT− ◦ T+ ◦ T−g = T−g and T+ ◦ T− ◦ T+f = T+f.In particular,T (µ, ν) = sup{∫YT+ ◦ T−g(y) dν(y)−∫XT−g dµ(x); g ∈ C(Y )}= sup{∫YT+f(y) dν(y)−∫XT− ◦ T+f dµ(x); f ∈ C(X)}.Proof. We can write for any g ∈ C(Y ),T−g(x) = supσ∈P(Y ){∫Ygdσ − T (δx, σ)}Since by assumption T+f ∈ C(Y ), we have with g = T+f ,T− ◦ T+f(x) = supσ∈P(Y ){∫YT+fdσ − T (δx, σ)}.At the same time, we have∫Y T+fdσ = infµ∈P(X){∫X fdµ+ T (µ, σ)} sinceT is a forward linear transfer. Substituting into the above yieldsT− ◦ T+f(x) = supσ∈P(Y )infµ∈P(X){∫Xfdµ+ T (µ, σ)− T (δx, σ)}≤ f(x).By a similar argument, we also deduce that T+ ◦ T−g(y) ≥ g(y) for all115g ∈ C(Y ).By replacing f ∈ C(X) in the inequality T− ◦T+f(x) ≤ f(x) with T−g,we obtain, T− ◦ T+ ◦ T−g ≤ T−g. At the same time applying T− to theinequality T+ ◦ T−g ≥ g, implies T− ◦ T+ ◦ T−g ≥ T−g. Therefore we haveequality: T− ◦ T+ ◦ T−g = T−g. Similarly, we have T+ ◦ T− ◦ T+f = T+f .Finally, since T (µ, ν) = supg∈C(Y ){∫Y gdν −∫X T−gdµ}, we can replaceg with T+ ◦ T−g ∈ C(Y ) in the supremum since T+ ◦ T−g ≥ g.5.2 Man˜e´ constant and weak KAM solutions forbackward linear transfersIn this section, we introduce the analogous notions of subsolutions, the Man`e´constant, and weak KAM solutions of Section 4.3.1. Throughout this sec-tion, let T : P(X)×P(X)→ R∪ {+∞} be a backward linear transfer suchthat {δx ; x ∈ X} ⊂ D1(T ) and assume c(T ) := infµ∈P(X) T (µ, µ) < +∞.Note that c(T ) > −∞ by the standing assumption that T is bounded be-low (recall this is part of the definition of a backward linear transfer). Itis always possible to assume without loss of generality that c(T ) = 0 bysimply considering the transfer T − c(T ); however we prefer to explicitlywrite the constant in the following. In addition, by compactness and lowersemi-continuity, there always exists at least one minimizer µ ∈ P(X) so thatT (µ, µ) = c(T ).We first begin by showing that c(T ) can be expressed in alternative ways.Proposition 5.2.1. Let T : P(X) × P(X) → R ∪ {+∞} be a backwardlinear transfer such that c(T ) := infµ∈P(X) T (µ, µ) < +∞. Thenc(T ) = limn→∞inf(µ,ν) Tn(µ, ν)n.Proof. First note thatinf(µ,ν)Tn(µ, ν) ≤ Tn(µ, µ) ≤ nT (µ, µ)116hencelim supn→∞inf(µ,ν) Tn(µ, ν)n≤ infµT (µ, µ) = c(T ).On the other hand, let µn1 , µnn+1 be such thatinf(µ,ν)Tn(µ, ν) = Tn(µn1 , µnn+1). (5.6)By definition of Tn as an inf-convolution of T with itself n-times, we maywriteTn(µn1 , µnn+1) =n∑j=1T (µnj , µnj+1)for some µnj ∈ P(X), j = 1, . . . , n + 1 (the infimum is achieved by weak∗lower semi-continuity). Hence by joint convexity,1nTn(µn1 , µnn+1) =1nn∑j=1T (µnj , µnj+1) ≥ T (1nn∑j=1µnj ,1nn∑j=1µnj+1). (5.7)Define νn :=1n∑nj=1 µnj . ThenT ( 1nn∑j=1µnj ,1nn∑j=1µnj+1) = T (νn, νn +1n(µnn+1 − µn1 )) (5.8)Now let nk be a subsequence such thatlim infn→∞inf(µ,ν) Tn(µ, ν)n= limk→∞inf(µ,ν) Tnk(µ, ν)nk.Up to extracting a further subsequence, we may assume that νnk → ν¯ forsome ν¯ ∈ P(X). It then follows from (5.6), (5.7), and (5.8), together withweak∗ lower semi-continuity of T , thatlim infn→∞1ninf(µ,ν)Tn(µ, ν) ≥ T (ν¯, ν¯) ≥ c(T ),which concludes the proof.Proposition 5.2.2. If T is a backward linear transfer on P(X) × P(X)117such that supx∈X infν∈P(X) T (δx, ν) < +∞, thenc(T ) = supg∈C(X)infx∈X{g(x)− T−g(x)}.Proof. If supx∈X infν∈P(X) T (δx, ν) < +∞, then by Lemma 5.1.1, T−g ∈USCb(X) for all g ∈ C(X). Consider now the functionf : P(X)× C(X)→ R, f(µ, g) :=∫X(g − T−g)dµ.Note that f is real-valued since T−g ∈ USCb(X). We have that g 7→ f(µ, g)is upper semi-continuous on C(X) since T− is convex and weakly lowersemi-continuous on C(X), and µ 7→ f(µ, g) is lower semi-continuous onP(X) since T−g ∈ USCb(X). Moreover, µ 7→ f(µ, g) is quasi-convex, i.e.{µ ∈ P(X) ; f(µ, g) ≤ λ} is convex or empty for λ ∈ R, and g 7→ f(µ, g) isquasi-concave, i.e. {g ∈ C(X) ; f(µ, g) ≥ λ} is convex or empty for λ ∈ R.Therefore by Sion’s minimax theorem (see Theorem 2.7.6), we haveinfµ∈P(X)supg∈C(X)f(µ, g) = supg∈C(X)infµ∈P(X)f(µ, g).Therefore we havec(T ) = infµ∈P(X)T (µ, µ) = supg∈C(X){∫X(g − T−g)dµ}= infµ∈P(X)supg∈C(X)f(µ, g)= supg∈C(X)infµ∈P(X)f(µ, g)= supg∈C(X)infµ∈P(X){∫X(g − T−g)dµ}.Since g−T−g is a lower semi-continuous function bounded below, it achievesa minimum on the compact space X, so that infµ∈P(X){∫X(g − T−g)dµ} =118minx∈X{g(x)− T−g(x)}. Consequently,c(T ) = supg∈C(X)infx∈X{g(x)− T−g(x)}.Definition 5.2.3. Let k ∈ R. A function g ∈ USC(X) is a subsolutionfor T− at level k iff1. T−g(x) + k ≤ g(x) for all x ∈ X, and2.∫X gdµ > −∞ for some µ ∈ P(X) which achieves the minimumT (µ, µ) = c(T ).Regarding the role of item 2 in the definition of a subsolution, we wishto be able to compare the constant k with c(T ) (see below in this section);note that by definition of c(T ), we have ∫X gdµ ≤ ∫X T−gdµ + c(T ). Thecondition is essentially telling us that a subsolution must be proper in aparticular way: that it is finite on the support of some minimising measureµ.Lemma 5.2.4. For every k < supg∈C(X) infx∈X{g(x)−T−g(x)}, there existsa subsolution g ∈ C(X).Proof. This is immediate, since by definition, there exists a sequence (gj) ⊂C(X) such thatinfx∈X{gj(x)− T−gj(x)} ↗ supg∈C(X)infx∈X{g(x)− T−g(x)}.Definition 5.2.5. The Man˜e´ constant c0 is the supremum over all k ∈ Rsuch that there exists a subsolution g for T− at level k.Lemma 5.2.6. The Man˜e´ constant c0 satisfiessupg∈C(X)infx∈X{g(x)− T−g(x)} ≤ c0 ≤ c(T ).119In particular, if supx∈X infν∈P(X) T (δx, ν) < +∞, then c0 = c(T ).Proof. It is immediate from the previous lemma thatc0 ≥ supg∈C(X)infx∈X{g(x)− T−g(x)}since there is a subsolution at every level k strictly below this value. Onthe other hand, take µ ∈ P(X) for which c(T ) = T (µ, µ). Then fromT (µ, µ) = suph∈C(X){∫X hdµ−∫X T−hdµ}, we have for any h ∈ C(X),∫X(h− T−h)dµ ≤ c(T ).For any g ∈ USC(X), take a decreasing sequence (hj) ⊂ C(X) with hj ↘ g.Then by Lemma 5.1.3, we have T−hj ↘ T−g, so by monotone convergence∫Xgdµ ≤∫XT−gdµ+ c(T ).Therefore for any level k for which there exists g ∈ USC(X) which is asubsolution, we have∫Xgdµ ≤∫XT−gdµ+ c(T ) ≤∫Xgdµ+ c(T )− khence since∫X gdµ > −∞, we may subtract from both sides and deducek ≤ c(T ). Consequently c0 ≤ c(T ).Definition 5.2.7. A function g ∈ USC(X) is a backward weak KAMsolution at level k for T if1. T−g(x) + k = g(x) for all x ∈ X, and2.∫X gdµ > −∞ for some µ such that T (µ, µ) = c(T ).Proposition 5.2.8. If g ∈ USC(X) is a backward weak KAM solution atlevel k, then necessarily, k = c(T ).120Proof. Clearly, if g is a backward weak KAM solution at level k, then itis a subsolution, so k ≤ c0 ≤ c(T ). On the other hand, g also satisfiesT−n g + nk = g, where T−n is the n-fold composition of T− with itself. Thismeans that∫Xgdµ−nk =∫XT−n gdµ = supν∈P(X){∫Xgdν−Tn(µ, ν)} ≤ supx∈Xg(x)− inf(µ,ν)Tn(µ, ν)where for the second equality in the above, we have used Corollary 5.1.7.Dividing by n and letting n→∞, we have− k ≤ − limn→∞inf(µ,ν) Tn(µ, ν)n= −c(T ) (5.9)where the latter equality is by Proposition 5.2.1. The above inequality (5.9)then implies c(T ) ≤ k, which concludes the proof.We shall therefore say g is a backward weak KAM solution if and only ifg is a backward weak KAM solution at level c(T ).Corollary 5.2.9. If there exists a backward weak KAM solution g, then theMan˜e´ constant c0 is equal to c(T ).5.3 Idempotent linear transfersWe will in subsequent sections construct an idempotent backward Kan-torovich operator that maps into the set of backward weak KAM solutions;its induced backward linear transfer will be idempotent. Therefore in thissection, we study idempotent backward linear transfers.Definition 5.3.1. Let T : P(X)×P(X)→ R∪{+∞} be a given functional.1. We say T is idempotent ifT (µ, ν) = T ? T (µ, ν) for all µ, ν ∈ P(X)whereT ? T (µ, ν) := inf{T (µ, σ) + T (σ, ν) ; σ ∈ P(X)}.1212. DefineA := {µ ∈ P(X) ; T (µ, µ) = 0}. We say that T isA-factorisable,if A 6= ∅, and T satisfiesT (µ, ν) = inf{T (µ, σ) + T (σ, ν) ; σ ∈ A} for all µ, ν ∈ P(X).3. T is distance-like, if it satisfiesT (µ, ν) ≤ T (µ, σ) + T (σ, ν), for all µ, σ, ν ∈ P(X).The following lemma and proposition is based on ([6], Lemma 11) wherewe weaken the continuity hypothesis to lower semi-continuity.Lemma 5.3.2. Let T : P(X) × P(X) → R ∪ {+∞} be weak∗ lower semi-continuous and idempotent. Fix µ, ν ∈ P(X). Let σ1, σ2 be such thatT (µ, ν) = T (µ, σ1) + T (σ1, ν) and T (σ1, ν) = T (σ1, σ2) + T (σ2, ν).Then T (µ, σ1) + T (σ1, σ2) = T (µ, σ2).Proof. This simply a consequence of T being idempotent. IndeedT (µ, ν) = T (µ, σ1) + T (σ1, σ2) + T (σ2, ν)≥ T ? T (µ, σ2) + T (σ2, ν)= T (µ, σ2) + T (σ2, ν)≥ T ? T (µ, ν)= T (µ, ν)so all the inequalities are in fact equalities. This means in particular com-paring the first and third line that T (µ, σ1) + T (σ1, σ2) = T (µ, σ2).Proposition 5.3.3. Let T : P(X)×P(X)→ R∪{+∞} be weak∗ lower semi-continuous and idempotent. Then T is distance-like and A-factorisable.Proof. It is immediate that T is distance-like if it is idempotent; indeed,122this is fromT (µ, ν) = T ? T (µ, ν) = infσ∈P(X){T (µ, σ) + T (σ, ν)}≤ T (µ, σ) + T (σ, ν) for all µ, ν, σ.Fix now µ, ν ∈ P(X). From T = T ? T , there exists σ1 ∈ P(X) such thatT (µ, ν) = T (µ, σ1) + T (σ1, ν); (5.10)similarly, there exists a σ2 such thatT (σ1, ν) = T (σ1, σ2) + T (σ2, ν). (5.11)We conclude from Lemma 5.3.2 thatT (µ, σ1) + T (σ1, σ2) = T (µ, σ2). (5.12)In addition, we if we add (5.10) and (5.11), we haveT (µ, ν) = T (µ, σ1) + T (σ1, σ2) + T (σ2, ν). (5.13)We therefore have (σ1, σ2) such that (5.10), (5.11), and (5.12) (and,consequently, (5.13)) hold.Continue this process with T (σ2, ν), to find a σ3 satisfying T (σ2, ν) =T (σ2, σ3) + T (σ3, ν). In this way we inductively create a sequence (σk)k∈Nwith the property that for (σ1, σ2, . . . , σk), we haveT (µ, ν) = T (µ, σ1) +k−1∑i=1T (σi, σi+1) + T (σk, ν) (5.14)and alsoT (µ, σ1) + T (σ1, σ2) = T (µ, σ2) (5.15)T (σk−1, σk) + T (σk, ν) = T (σk−1, ν), (5.16)123as well asm∑i=`T (σi, σi+1) = T (σ`, σm+1) (5.17)whenever 1 ≤ ` < m ≤ k − 1.In particular, the same properties (5.14), (5.15), (5.16), (5.17) above holdif we take instead m+1 terms (σk1 , . . . , σkm+1) of any increasing subsequencekj of k. In particular, this meansT (µ, σk1) +m∑j=1T (σkj , σkj+1) + T (σkm+1 , ν) = T (µ, ν). (5.18)We take now a subsequence σkj of σk converging to some σ ∈ P(X). Byweak∗ lower semi-continuity of T , we havelim infj→∞T (σkj , σkj+1) ≥ T (σ, σ)lim infj→∞T (µ, σkj ) ≥ T (µ, σ)lim infj→∞T (σkj , ν) ≥ T (σ, ν).In particular, given  > 0, for all but finitely many j, it must hold thatT (σkj , σkj+1) ≥ T (σ, σ)−  (5.19)T (µ, σkj ) ≥ T (µ, σ)−  (5.20)T (σkj , ν) ≥ T (σ, ν)−  (5.21)so up to removing the first N terms for a finite N = N, we may assume wehave a subsequence σkj satisfying (5.19), (5.20), and (5.21) for all j, as wellas (5.18) .Applying the inequalities of (5.19), (5.20), and (5.21), to (5.18), we ob-tainT (µ, ν) ≥ T (µ, σ) +mT (σ, σ) + T (σ, ν)− (m+ 2)for m ≥ 1. From the fact that T = T ? T , we have T (µ, σ) + T (σ, ν) ≥124T ? T (µ, ν) = T (µ, ν), so the above inequality impliesT (σ, σ) ≤ m+ 2m ≤ 2.As  > 0 is arbitrary, we obtain T (σ, σ) ≤ 0, and consequently T (σ, σ) = 0(the reverse inequality following from T = T ? T ).Finally, we note that T (µ, ν) = T (µ, σkj ) + T (σkj , ν) for all j, so at thelim inf, we findT (µ, ν) ≥ T (µ, σ) + T (σ, ν).The reverse inequality is immediate from T = T ? T .Definition 5.3.4. Suppose T− : USC(X) → USC(X) is some given map.We say that T− is idempotent if T− ◦ T−g = T−g for all g ∈ C(X).Remark 5.3.5. We observe that if T− : USC(X)→ USC(X) is an idem-potent backward Kantorovich operator, then the induced backward lineartransfer T (µ, ν) := supg∈C(X){∫X gdν−∫X T−gdµ} is idempotent thanks toCorollary 5.1.7.5.3.1 Examples of idempotent linear transfersWe present some examples of idempotent linear transfers.Example 5.3.6 (Convex energy). Recall the convex energy example ofSection 2.5.1. If I is any bounded below, convex, and weak∗ lower semi-continuous functional, and m := inf{I(σ);σ ∈ P(Y )}, then T (µ, ν) :=I(ν) − m is an idempotent backward linear transfer with an idempotentKantorovich map T−g := I∗(g) +m. Indeed, note thatT− ◦ T−g = I∗(T−g) +m = supν∈P(Y ){∫YT−gdν − I(ν)}+m= supν∈P(Y ){∫Y(I∗(g) +m)dν − I(ν)}+m= I∗(g) + 2m− infν∈P(Y )I(ν)= I∗(g) +m = T−g.125(See also Example 5.5.5 later in Section 5.5.3.)Example 5.3.7 (Markov operator). Any transfer induced by a boundedpositive linear operator T with T 2 = T and T1 = 1, and in particular, anypoint transformation σ such that σ2 = σ as in Example 2.5.7.Example 5.3.8 (Balayage transfer). The balayage transfer B of Example2.5.3 is idempotent since its Kantorovich map is T−g = gˆ, where, for exam-ple, in the case of balayage with convex functions, gˆ is the concave envelopeof g.Example 5.3.9 (Optimal Mass Transport). If Tc is an optimal mass trans-port associated to a bounded below lower semi-continuous cost functionc : X×X → R∪{+∞}, then Tc is idempotent if c(x, x) = 0 for every x ∈ Xand c satisfies the triangular inequalityc(x, z) ≤ c(x, y) + c(y, z) for all x, y, z in X. (5.22)In particular, this impliesTc(µ, ν) = sup{∫XT−c g d(ν − µ); g ∈ C(X)}. (5.23)For example, if c(x, y) = dX(x, y)p for 0 < p ≤ 1, then the correspondingoptimal mass transport is idempotent since c satisfies the reverse triangleinequality, and therefore c ? c(x, y) = infz∈X{c(x, z) + c(z, y)} = c(x, y).Example 5.3.10 ( An idempotent optimal Skorohod embedding). The fol-lowing transfer was considered in Ghoussoub-Kim-Palmer [31].T (µ, ν) := inf{E[ ∫ τ0L(t, Bt)dt]; τ ∈ S(µ, ν)}, (5.24)where S(µ, ν) denotes the set of (possibly randomized) stopping times withfinite expectation such that ν is realized by the distribution of Bτ (i.e.,Bτ ∼ ν in our notation), where Bt is Brownian motion starting with µ asa source distribution, i.e., B0 ∼ µ. Note that T (µ, ν) = +∞ if S(µ, ν) = ∅,which is the case if and only if µ and ν are not in subharmonic order. In this126case, it has been proved in [31] that under suitable conditions, the backwardlinear transfer is given by T−g = Jg(0, ·), where Jg : R+×Rd → R is definedvia the dynamic programming principleJg(t, x) := supτ∈Rt,x{Et,x[g(Bτ )−∫ τtL(s,Bs)ds]}, (5.25)where the expectation superscripted with t, x is with respect to the Brow-nian motions satisfying Bt = x, and the minimization is over all finite-expectation stopping timesRt,x on this restricted probability space such thatτ ≥ t. Jg(t, x) is actually a “variational solution” for the quasi-variationalHamilton-Jacobi-Bellman equation:min{J(t, x)− ψ(x)− ∂∂tJ(t, x)− 12∆J(t, x) + L(t, x)}= 0. (5.26)Note that Jg(t, x) ≥ g(x), that is T−g ≥ g for every g, hence (T−)2g ≥ T−g.Assume now t → L(t, x) is decreasing, which yields that t → J(t, x) isincreasing (see [31]). Given  > 0, fix τ such that(T−)2g(x) = JT−g(0, x) ≤ Et,x[T−g(Bτ)−∫ τtL(s,Bs)ds]+ ,hence since T−g(Bτ) = Jg(0, Bτ) ≤ Jg(t, Bτ),(T−)2g(x) ≤ Et,x[Jg(0, Bτ)−∫ τtL(s,Bs)ds]+ ≤ Et,x[Jg(t, Bτ)−∫ τtL(s,Bs)ds]+ ≤ Jg(0, x) +  = T−g(x) + ,where the last inequality uses the supermartingale property of the processt→ Jg(t, Bτ)−∫ τt L(s,Bs)ds. It follows that we obtain the reverse inequal-ity (T−)2g ≤ T−g, and T− is therefore idempotent.1275.3.2 T -Lipschitz functionalsProposition 5.3.11. Suppose T : P(X) × P(X) → R ∪ {+∞} is a weak∗lower semi-continuous, bounded below, jointly convex functional such that Tis A-factorisable. Then the following hold:1. For any bounded functional Φ : A → R that is T -Lipschitz, that is, forany Φ satisfyingΦ(ν)− Φ(µ) ≤ T (µ, ν), for all µ, ν ∈ A,there exists f ∈ C(X) such thatΦ(µ) =∫Xfdµ for all µ ∈ A.2. If T is, in addition, a backward linear transfer, andsupν∈P(X)infµ∈AT (µ, ν) < +∞,thenΦ(µ) =∫XT−fdµ =∫Xfdµ for every µ ∈ A. (5.27)3. If T is, in addition, both a forward and backward linear transfer, thenΦ(µ) =∫Xfdµ =∫XT−fdµ =∫XT+ ◦ T−fdµ for every µ ∈ A.Moreover, the functions ψ0 := T−f and ψ1 := T+ ◦T−f are conjugatein the sense that ψ0 = T−ψ1 and ψ1 = T+ψ0.4. Moreover, if g is a function in C(X) such that∫X gdµ = Φ(µ) for allµ ∈ A, thenψ0 ≤ T−g and ψ1 ≥ T+g. (5.28)128Proof. 1. Let Φ be such that µ→ Φ(µ) is T -Lipschitz on A and defineΦ0(µ) := supσ∈A{Φ(σ)− T (µ, σ)} and Φ1(ν) := infσ∈A{Φ(σ) + T (σ, ν)}.Note that Φ0 and Φ1 are both finite on A, but in general Φ0 may be −∞(resp., Φ1 may be +∞) on certain subsets of P(X), depending on the effec-tive domain of T .We now show that Φ0 ≤ Φ1 on P(X). This is trivally true if one of Φ0,Φ1 is not finite, so assume µ ∈ P(X) is such that Φ0(µ),Φ1(µ) ∈ R. Then,by definition of Φ0, Φ1, and the fact that Φ is T -Lipschitz on A, and T isA-factorisable, we may writeΦ0(µ)− Φ1(µ) = supσ,τ∈A{Φ(σ)− T (µ, σ)− Φ(τ)− T (τ, µ)}≤ supσ,τ∈A{Φ(σ)− Φ(τ)− T (τ, σ)}≤ 0.Note now that Φ0 is concave and weak∗ upper semi-continuous, while Φ1 isconvex and weak∗ lower semi-continuous, on the convex subset P(X) of thereal vector spaceM(X). Thus by Hahn-Banach (see, e.g. [50] p.319), thereexists f¯ ∈ C(X) such that µ 7→ ∫X fdµ on P(X) lies between Φ0 and Φ1:Φ0(µ) ≤∫Xfdµ ≤ Φ1(µ) for all µ ∈ P(X). (5.29)On the other hand, it also holds by the definition of Φ0 and Φ1, thatΦ1(µ) ≤ Φ(µ) ≤ Φ0(µ) for all µ ∈ A (5.30)so that (5.29) and (5.30) together showsΦ(µ) = Φ1(µ) = Φ0(µ) =∫Xfdµ for all µ ∈ A.2. Suppose now T is additionally a backward linear transfer with T− as129its Kantorovich operator. If µ ∈ A,∫XT−fdµ = supν∈P(X){∫Xfdν − T (µ, ν)} ≥∫Xfdµ− T (µ, µ) =∫Xfdµ.Assume now that supν∈P(X) infµ∈A T (µ, ν) < +∞. It can be checked thatthis implies Φ0, Φ1 are bounded. Then it holds thatsupν∈P(X){Φ1(ν)− T (µ, ν)} ≤ supν∈P(X)Φ1(ν)− C < +∞where C is a lower bound for T . This means that for any µ ∈ A, by (5.29),we can write−∞ <∫XT−fdµ = supν∈P(X){∫Xfdν−T (µ, ν)} ≤ supν∈P(X){Φ1(ν)−T (µ, ν)}.(5.31)In view of item 1, we will be done if we show that we have the conjugateformula,supσ∈P(X){Φ1(σ)− T (µ, σ)} = Φ0(µ). (5.32)To this end, for every µ ∈ P(X), we haveΦ0(µ) = supσ∈A{Φ(σ)− T (µ, σ)} = supσ∈A{Φ1(σ)− T (µ, σ)} ≤ supσ∈P(X){Φ1(σ)− T (µ, σ)}.On the other hand, for any ν, µ ∈ P(X), we haveΦ1(ν)− Φ0(µ) = infσ,τ∈A{Φ(σ) + T (σ, ν)− Φ(τ) + T (µ, τ)}≤ infσ,τ∈A{T (σ, ν) + T (τ, σ) + T (µ, τ)}≤ infσ∈A{T (σ, ν) + T (µ, σ)}= T (µ, ν).This shows (5.32).3. Suppose in addition that T is a forward linear transfer with T+ as a130Kantorovich operator. Recalling the property T+ ◦ T−f ≥ f , we have∫XT+ ◦ T−fdµ ≥∫Xfdµ ≥ Φ0(µ).On the other hand, by (5.31) and (5.32),∫XT+ ◦T−fdµ = infσ∈P(X){∫XT−fdσ+T (σ, µ)} ≤ infσ∈P(X){Φ0(σ)+T (σ, µ)}.In other words, T−f and T+ ◦ T−f are two conjugate functions verifying∫XT+ ◦ T−fdµ =∫XT−fdµ = Φ(µ) for all µ ∈ A.4) To prove (5.28), first note that∫XT−fdµ ≤ Φ0(µ) = sup{Φ(σ)− T (µ, σ);σ ∈ A}≤ sup{∫Xgdσ − T (µ, σ);σ ∈ P(X)}=∫XT−gdµ.On the other hand,∫XT+ ◦ T−fdµ = inf{∫XT−fdσ + T (σ, µ);σ ∈ P(X)}= inf{∫XT−fdσ + T (σ, λ) + T (λ, µ);λ ∈ A, σ ∈ P(X)}= inf{∫XT+ ◦ T−fdλ+ T (λ, µ);λ ∈ A}= inf{∫Xgdλ+ T (λ, µ);λ ∈ A}≥ inf{∫Xgdλ+ T (λ, µ);λ ∈ P(X)}=∫XT+gdµ,which completes the proof.131Corollary 5.3.12. For every idempotent backward linear transfer T suchthat {δx ; x ∈ X} ⊂ A and supν∈P(X) infµ∈A T (µ, ν) < +∞, there is afunction f ∈ C(X) which is fixed by T−, i.e. T−f(x) = f(x).5.4 Ergodic properties of equicontinuoussemigroups of transfersWe now proceed to the construction of the backward weak KAM solutionsfirst introduced in Section 5.2. Given a backward linear transfer T withbackward Kantorovich operator T−, we shall construct a backward Kan-torovich operator T−∞ which maps C(X) into the set of backward weakKAM solutions for T−. This operator will be idempotent, and its inducedbackward linear transfer T∞ is the analog of the Peierls barrier of standardAubry-Mather theory in Chapter 4.We begin by assuming T is weak∗ continuous on P(X) × P(X). Weuse the notation Tn to denote, for each n ∈ N, the inf-convolution Tn =T ? T ? .... ? T n-times, and T−n the n-fold composition of T− with itself.Recall that the quadratic Wasserstein distance, denoted by W2(µ, ν) :=√T2(µ, ν) where T2 is optimal transport with quadratic cost c(x, x′) :=dX(x, x′)2, metrizes the topology of weak∗ convergence on P(X) when Xis compact. Therefore, if T is weak∗ continuous on P(X) × P(X), thenit is weak∗ uniformly continuous since P(X) × P(X) is weak∗ compact.Therefore, there exists a modulus of continuity δ : [0,∞)→ [0,∞), δ(0) = 0,such that|T (µ, ν)− T (µ′, ν ′)| ≤ δ(W2(µ, µ′) +W2(ν, ν ′)) for all µ, µ′, ν, ν ′ ∈ P(X).We start with the following lemma which can be found in Bernard-Buffoni [6] and adapted to our setting at hand.Lemma 5.4.1 (Bernard-Buffoni [6], Lemma 9). Let T : P(X)×P(X)→ Rbe a weak∗ continuous functional with a modulus of continuity δ. Then thefamily {Tn}n∈N are equicontinuous with the same modulus of continuity δ,and there exists a positive constant C > 0 such that132|Tn(µ, ν)− nc(T )| ≤ C, for every n ≥ 1 and all µ, ν ∈ P(X),where we recall from Section 5.2 thatc(T ) = infµT (µ, µ) = limn→∞inf{Tn(µ, ν) ; µ, ν ∈ P(X)}n.Proof. Define Mn := maxµ,ν Tn(µ, ν) and M := infn≥1{Mnn } > −∞. Thesequence {Mn}n≥1 is subadditive, that isMn+m ≤Mn+Mm, hence {Mnn }n≥1decreases to its infimum M as n→∞. Indeed (see e.g. [9] Lemma 1.18), fixn > 0 and write for any m, the decomposition m = nq+r, where 0 ≤ r < n.The subadditivity of Mn impliesMmm=Mnq+rnq + r≤ Mnqnq+Mrnq≤ Mnn+Mrnq.We therefore obtain lim supm→∞Mmm ≤ Mnn . On the other hand, infn≥1 Mnn ≤lim infn→∞ Mnn . Therefore,Mnn converges to M as n→∞.In addition, if mn := minµ,ν Tn(µ, ν), then the above applied to −mn yieldsthat limn→∞ mnn = m, where m := supnmnn .We now show that m = M . Note that the family Tn all have the samemodulus of continuity; this follows exactly because (µ, ν) 7→ T (µ, σ1) +T (σ1, σ2) + . . . + T (σn, ν) has modulus of continuity δ for each choice of(σ1, . . . , σn) ∈ P(X)× . . .×P(X), so the infimum (i.e. the function Tn) alsohas modulus of continuity δ. The uniform modulus of continuity δ impliesthe existence of a constant C > 0, such that Mn −mn ≤ C for every n ≥ 1.Then, we obtain the string of inequalitiesnM − C ≤Mn − C ≤ mn ≤ Tn(µ, ν) ≤Mn ≤ mn + C ≤ nm+ C.The left-most and right-most inequalities imply M ≤ m upon sending n→∞, hence m = M .Lemma 5.4.2. 1. For any g ∈ C(X), there is a constant C > 0 such133that|T−n g(x) + nc(T )− supXg| ≤ C for all n ∈ N and all x ∈ X.2. The semi-group of operators {T−n }n≥1 has the same modulus of conti-nuity as T .3. The constant c(T ) is critical in the sense that T−n g + kn → ±∞ asn → ∞ if k 6= c(T ), depending on if k < c(T ) or k > c(T ), for anyg ∈ C(X).Proof. 1) By Lemma 5.4.1 and since T−n g(x) + nc(T ) = supσ{∫gdσ −(Tn(δx, σ)− nc(T ))}, we have supX(g)−C ≤ T−n g(x)+nc(T ) ≤ supX(g)+C.For 2) we note thatT−n g(x) = supσ{∫gdσ − Tn(δx, σ)}≤ supσ{∫gdσ − Tn(δy, σ)}+ supσ{Tn(δy, σ)− Tn(δx, σ)}= T−n g(y) + δ(d(x, y)).We now interchange x and y to obtain the reverse inequality.3) follows from 1) sincesupX(g)− C + (k − c)n ≤ T−n g(x) + kn ≤ supX(g) + C + (k − c)n.We shall prove the following, which is an analogue of Bernard-Buffoni[5, 6].Theorem 5.4.3 (Ergodic properties of weak∗ continuous linear transfers).Let T be a weak∗ continuous backward linear transfer on M(X) ×M(X)with modulus of continuity δ, and with backward Kantorovich operator T− :C(X) → C(X). Then, there exists a backward Kantorovich operator T−∞ :134C(X)→ C(X), which may be taken to beT−∞g(x) := limn→∞(T−n ◦ T−g(x) + nc(T ))where T−g(x) := lim supn→∞(T−n g(x) + nc(T )). Together with its corre-sponding backward linear transfer T∞, they satisfy:1. T−∞ maps every g ∈ C(X) to a backward weak KAM solution for T−,i.e.,T−n ◦ T−∞g + nc(T ) = T−∞g and T−∞ ◦ T−n g + nc(T ) = T−∞gfor all n ∈ N, where c(T ) = infµ T (µ, µ) = limn→∞ inf{Tn(µ,ν) ;µ,ν∈P(X)}n .2. T−∞ is idempotent, i.e. T−∞ ◦ T−∞g = T−∞g for all g ∈ C(X).3. T∞ satisfies,(Tn−nc(T ))?T∞(µ, ν) = T∞(µ, ν) = T∞?(Tn−nc(T ))(µ, ν) for every n ∈ N.4. T∞ is idempotent and therefore A-factorisable, i.e. the set A := {σ ∈P(X); T∞(σ, σ) = 0} is non-empty, and for every µ, ν ∈ P(X), wehaveT∞(µ, ν) = inf{T∞(µ, σ) + T∞(σ, ν), σ ∈ A},and the infimum on A is attained.5. For every µ, ν ∈ P(X), we havesup{∫T−∞gd(ν − µ) ; g ∈ C(X)}≤ T∞(µ, ν) ≤ lim infn→∞ (Tn(µ, ν)−nc(T )).6. If T (µ, µ) = c(T ), then µ ∈ A. Additionally, (µ, µ) ∈ D if and only ifµ ∈ A and T (µ, µ) = c(T ), whereD := {(µ, ν) ∈ P(X)× P(X) : T (µ, ν) + T∞(ν, µ) = c(T )}.135Remark 5.4.4. The backward linear transfer T∞ is an analog of the Peierlsbarrier.Proof. 1) Given g ∈ C(X), defineT−g(x) := lim supn→∞(T−n g(x) + nc(T )).The first observation we make is that T−is a backward Kantorovich oper-ator. Indeed, note that the monotonicity, convexity, and affine with respectto constants, properties of T− are inherited by T−. The remaining propertyto check is the lower semi-continuity. For this, suppose gk → g in C(X) forthe sup norm. We have by Proposition 2.4.4 thatT−n g(x) ≤ T−n gk(x) + ‖g − gk‖∞,so thatT−g(x) ≤ T−gk(x) + ‖g − gk‖∞where k →∞ then yields T−g(x) ≤ lim infk→∞ T−gk(x). By Lemma 5.4.2,T−g is continuous and has a modulus of continuity δ, and ‖T−g‖∞ ≤supX(g) + C.Now the sequence supm≥n{T−mg + mc(T )} is a sequence of continuousfunctions that decrease monotonically to T−g pointwise as n→∞, we mayapply Lemma 5.1.3 to deduce,T− ◦ T−g(x) = limn→∞T− ◦[supm≥n{T−mg + c(T )m}](x)≥ limn→∞ supm≥n{T−m+1g(x) + c(T )m}= limn→∞ supm≥n{T−m+1g(x) + (m+ 1)c(T )} − c(T )= T−g(x)− c(T ).Therefore, T− ◦ T−g(x) + c(T ) ≥ T−g(x). By monotonicity of the operator136T−, this inequality impliesT−n ◦ T−g(x) + nc(T ) ≥ T−m ◦ T−g(x) +mc(T )whenever n ≥ m, i.e. {T−n ◦ T−g + nc(T )}n≥1 is a monotone increasingsequence of continuous functions, with|T−n ◦ T−g(x)− T−n ◦ T−g(x′)| ≤ δ(dX(x, x′)).In addition, we have from Lemma 5.4.2 the uniform in n bound‖T−n ◦ T−g(x) + nc(T )‖∞ ≤ ‖T−g‖∞ + C ≤ ‖g‖∞ + 2C.We therefore have a monotone increasing sequence of equicontinuous func-tions {T−n ◦ T−g + nc(T )}n∈N bounded above uniformly in n and x. Ittherefore converges pointwise to a continuous function, and we can definethe pointwise limit via the operator T−∞ : C(X) → C(X) defined via theformula,T−∞g(x) := limn→∞T−n ◦ T−g(x) + nc(T ).We now claim that T−∞ is a backward Kantorovich operator. Indeed, sinceT−n and T−are backward Kantorovich operators, the monotonicity, convex-ity, and affine on constants properties hold in the pointwise limit n → ∞,and therefore hold for T−∞. Regarding lower semi-continuity, note that wecan repeat a similar argument as was given for T−to deduce thatT−n ◦ T−g(x) ≤ T−n ◦ T−gk + ‖g − gk‖∞. (5.33)which givesT−∞g(x) ≤ lim infk→∞T−∞gk(x).Since the convergence is monotone (even uniform, by Dini’s theorem) we137can apply Lemma 5.1.3 to deduce thatT− ◦ T−∞g(x) + c(T ) = limm→∞T− ◦[T−m ◦ T−g(x) +mc(T )]+ c(T )= limm→∞{T−m+1 ◦ T−g(x) + (m+ 1)c(T )}= T−∞g(x)so repeated application of T− yields T−n ◦ T−∞g(x) + nc(T ) = T−∞g(x).We can also consider the compositionT−∞ ◦ T−g(x) = limn→∞(T−n ◦ T− ◦ T−g(x) + nc(T )) (5.34)and we haveT−T−g(x) = lim supn→∞(T−n T−g(x) + nc(T ))= lim supn→∞(T−n+1g(x) + (n+ 1)c(T ))− c(T )= T−g(x)− c(T )which means from (5.34) thatT−∞ ◦ T−g(x) + c(T ) = T−∞g(x).2) From T−n ◦ T−∞g(x) + nc(T ) = T−∞g(x), we obtain T− ◦ T−∞g(x) =T−∞g(x) Consequently by definition of T−∞, this further implies that T−∞ ◦T−∞g(x) = T−∞g(x) so T−∞ is idempotent.3) T−∞ is a Kantorovich operator, thus we may defineT∞(µ, ν) := supg∈C(X){∫Xgdν −∫XT−∞gdµ}and it is a backward linear transfer; from T−n ◦ T−∞g + nc(T ) = T−∞g andT−∞ ◦ T−n g + nc(T ) = T−∞g, it therefore satisfiesT∞(µ, ν) = (Tn−nc(T ))?T∞(µ, ν) = T∞?(Tn−nc(T ))(µ, ν), for all n ≥ 1.1384) From T−∞ ◦ T−∞g(x) = T−∞g(x), it satisfiesT∞(µ, ν) = T∞ ? T∞(µ, ν), for all µ, νso T∞ is idempotent. Therefore by Proposition 5.3.3, T∞ is A-factorisable.5) Note from 1) that T−∞g(x) ≥ lim supn→∞(T−n g(x) + nc(T )), so∫XT−∞gdµ ≥∫Xlim supn→∞(T−n g(x) + nc(T ))dµ≥ lim supn→∞∫X(T−n g(x) + nc(T ))dµ.HenceT∞(µ, ν) ≤ sup lim infn→∞{∫Xgdν −∫XT−n gdµ− nc(T ) ; g ∈ C(X)}≤ lim infn→∞ sup{∫Xgdν −∫XT−n gdµ− nc(T ) ; g ∈ C(X)}= lim infn→∞ (Tn(µ, ν)− nc(T )).On the other hand, from T−∞ ◦ T−∞g = T−∞g,T∞(µ, ν) = sup{∫Xgdν −∫XT−∞gdµ ; g ∈ C(X)}≥ sup{∫XT−∞gd(ν − µ) ; g ∈ C(X)}.6) Suppose µ is a measure which realises c(T ) = T (µ, µ). Then by 6),we have0 ≤ T∞(µ, µ) ≤ lim infn→∞ (Tn(µ, µ)− nc(T ))≤ lim infn→∞ (nT (µ, µ)− nc(T )) = 0,so µ ∈ A. If (µ, µ) ∈ D, then we immediately see µ ∈ A and T (µ, µ) = c(T );the converse is also immediate.139Remark 5.4.5. The above theorem also holds when we have an appropriatesemi-group of backward linear transfers (Tt)t>0. In this case, we do not builda discrete-time semi-group (Tn)n∈N from one linear transfer T , but insteadalready have in hand an appropriate semi-group (Tt).In particular, if {Tt}t≥0 is a family of backward linear transfers onP(X) × P(X) with associated Kantorovich operators {Tt}t≥0, where T0 isthe identity transfer,T0(µ, ν) =0 if µ = ν ∈ P(X)+∞ otherwise.then under the following assumptions(H0) The family {Tt}t≥0 is a semi-group under inf-convolution: Tt+s = Tt?Tsfor all s, t ≥ 0.(H1) For every t > 0, the transfer Tt is weak∗-continuous, and the Diracmeasures are contained in D1(Tt).(H2) For any  > 0, {Tt}t≥ has common modulus of continuity δ (possiblydepending on ).the results of Theorem 5.4.3 and their proofs hold (with appropriate changesfrom n to t). For clarity we decided to present Theorem 5.4.3 for the discrete-time case.5.4.1 The case of optimal transport for continuous costfunctionsWe now identify T−∞ and T∞ associated to a semi-group of linear transferswhich are given by mass transports.Proposition 5.4.6. Suppose ct(x, y) is a semi-group of equicontinuous costfunctions on X ×X, that isct+s(x, y) = ct ? cs(x, y) := inf{ct(x, z) + cs(z, y); z ∈ X},140and suppose inf(x,y) ct(x, y) = infx ct(x, x). Without loss of generality, as-sume infx ct(x, x) = 0, and consider the associated optimal mass transportsTt(µ, ν) = inf{∫X×Xct(x, y)dpi(x, y) ; pi ∈ K(µ, ν)}.1. The family (Tt)t then forms a semi-group of linear transfers for theconvolution operation i.e., Tt+s = Tt ? Ts for any s, t ≥ 0 that isequicontinuous on P(X)×P(X), hence one can associate an idempo-tent Kantorovich operator T−∞ and associated backward linear transferT∞.2. The following holds for the constant c := limt→∞infµ,ν∈P(X) Tt(µ,µ)t :c = min{∫X×Xc1(x, y)dpi;pi ∈ P(X ×X), pi1 = pi2} (5.35)3. Letting c∞(x, y) := lim inft→∞(ct(x, y)− ct), then c∞ is continuous onP(X)× P(X), and:T∞(µ, ν) = Tc∞(µ, ν) := inf{∫X×Xc∞(x, y)dpi(x, y) ; pi ∈ K(µ, ν)},T−∞f(x) = sup{f(y)−c∞(x, y) ; y ∈ X} and T+∞f(y) = inf{f(x)+c∞(x, y) ; x ∈ X}4. The set A := {σ ∈ P(X); T∞(σ, σ) = 0} consists of those σ ∈ P(X)supported on the set A = {x ∈ X; c∞(x, x) = 0}.5. The minimizing measures in (5.35) are all supported on the setD := {(x, y) ∈ X ×X ; c1(x, y) + c∞(y, x) = c}.Proof. We apply Theorem 5.4.3 in this context: The Kantorovich operatorfor Tt is given by T−t g(x) = sup{g(y) − ct(x, y) ; y ∈ X} and as shownin Proposition 2.10.10, we have Ts+t = Tct?cs = Tct ? Tcs = Tt ? Ts, andT−t+s = T−t ◦T−s for every s, t. For the idempotent Kantorovich operator T−∞associated to (Tt)t, we recall that in the proof of Theorem 5.4.3 we had the141operatorT−g(x) := lim supt→∞(T−t g(x) + ct).We claim that T−g(x) = T−c∞g(x) where T−c∞g(x) := supy∈X{g(y)−c∞(x, y)}is the backward Kantorovich operator for the optimal transport with costc∞, so that in fact T−∞ = T−c∞ . Indeed, first note thatlim supt(T−t g(x) + ct) ≥ supy∈X{g(y)− c∞(x, y)} = T−c∞g(x). (5.36)On the other hand, let yn achieve the supremum for T−n g(x) = sup{g(y) −cn(x, y) ; y ∈ X}, and let (nj)j be a subsequence such that limj→∞(T−njg(x)+cnj) = lim supn(Tnf(x)+ cn). By refining to a further subsequence, we mayassume by compactness of X, that ynj → y¯ as j → ∞. Then by equi-continuity of the cn’s, we deduce thatlim supn(T−n g(x)+cn) = limj→∞(T−njg(x)+cnj) = g(y¯)− lim infj (cnj (x, y¯)−cnj).(5.37)As lim infj(cnj (x, y¯)− cnj) ≥ lim infn(cn(x, y¯)− cn) = c∞(x, y¯), we obtainlim supn(T−n g(x) + nc) ≤ g(y¯)− c∞(x, y¯) ≤ supy{g(y)− c∞(x, y)} = T−c∞g(x).(5.38)The inequality (5.38) is true for every sequence (nk)k going to ∞, so wededuce that lim supt(T−t g(x) + ct) ≤ T−c∞g(x), and hence combining thiswith (5.36) gives equality: lim supt(T−t g(x) + ct) = T−c∞g(x).Finally, we note that T−s (lim supt(T−t g+ct))(x)+cs = T−s T−c∞g(x)+cs =T−c∞g(x) thanks to the fact that cs?c∞ = c∞. This implies from the definitionof T−∞ as the limit as s→∞ (see Theorem 5.4.3) that T−∞g(x) = T−c∞g(x).Properties (1), (2) and (3) follow then immediately. Properties (4) and(5) now follow since ct(x, y) is minimised (by assumption) on the diagonalct(x, x), so that T∞(µ, µ) = 0 = Tc∞(µ, µ) implies µ is supported on setwhere c∞(x, x) = 0.Example 5.4.7 (Iterates of power costs). Let X ⊂ Rn be compact and142convex, and c(x, y) := |x − y|p for p > 0 and x, y ∈ X. Let Tc denote theoptimal transport with cost c, with corresponding backward Kantorovichoperator T−c g(x) = supy∈X{g(y)− c(x, y)}.If 0 < p ≤ 1, then c satisfies the reverse triangle inequality c(x, z) +c(z, y) ≥ c(x, y), hence c ? c(x, y) = inf{|x− z|p + |z− y|p ; z ∈ X} = c(x, y).Therefore T−∞g(x) = T−c g(x) (i.e. T−c is itself idempotent).If p > 1, then c ? c(x, y) = inf{|x − z|p + |z − y|p ; z ∈ X} is minimisedat some point z = (1 − λ)x + λy on the line between x and y, so thatcp ? cp(x, y) = (λp + (1− λ)p) |x− y|p. The optimal λ is 12 . Hence,(T−c )ng(x) = supy∈X{g(y)− 1np−1|x− y|p}.Therefore, when n → ∞, (T−c )ng(x) → supx∈X g(x), and it follows thatT−∞g(x) := supx∈X g(x), with the corresponding backward linear transferT∞ being the null transfer of Example 2.5.1.5.4.2 Aubry-Mather theory and weak KAM theory forLagrangian systemsIn this section, we briefly mention how, when the cost semi-group of theprevious section is generated from a Lagrangian, how Theorem 5.4.3 andProposition 5.4.6 fits in the context of the Aubry-Mather theory, weak KAM,and Bernard-Buffoni connection to optimal transport discussed in Chapter4.Let L be a time-independent Tonelli Lagrangian on M = Tn the flattorus, and consider Tt to be the cost minimizing transportTt(µ, ν) = inf{∫M×Mct(x, y)dpi(x, y) ; pi ∈ K(µ, ν)},wherect(x, y) := inf{∫ t0L(γ(s), γ˙(s))ds ; γ ∈ C1([0, t];M); γ(0) = x, γ(t) = y}.Recall from Chapter 4 (see Section 4.3), the Lax-Oleinik semi-group S−t ,143t > 0 defined by the formulaS−t u(x) := inf{u(γ(0)) +∫ t0L(γ(s), γ˙(s))ds ; γ ∈ C1([0, t];M), γ(t) = x},as well as the semi-groupS+t u(x) := sup{u(γ(t))−∫ t0L(γ(s), γ˙(s))ds ; γ ∈ C1([0, t];M), γ(0) = x}.Theorem 5.4.8. There exists a unique constant c ∈ R such that the follow-ing hold:1. (Fathi [20]) There exists a function u− : M → R (resp. u+) such thatS−t u− − ct = u− (resp. S+t u− + ct = u−) for each t ≥ 0.2. (Bernard-Buffoni [5]) Let c∞(x, y) := lim inft→∞ ct(x, y) denote thePeierls barrier function. The following duality then holds:inf{∫M×Mc∞(x, y)dpi(x, y) ; pi ∈ K(µ, ν)} = supu+,u−{∫Mu+dν−∫Mu−dµ},where the supremum ranges over all u+, u− ∈ C(M) such that u+(resp. u−) is a positive (resp. negative) weak KAM solution, and suchthat u+ = u− on the set A := {x ∈ M ; c∞(x, x) = 0}. Moreover,c∞(x, y) = minz∈A{c∞(x, z) + c∞(z, y)}.3. (Bernard-Buffoni [6]) The constant c satisfiesc = minpi∫M×Mc1(x, y)dpi(x, y),where the minimum is taken over all pi ∈ P(M ×M) with equal firstand second marginals. The minimizing measures are all supported onD := {(x, y) ∈M ×M ; c1(x, y) + c∞(y, x) = c}.4. (Mather [43]) The constant c = infm∫TM L(x, v)dm(x, v) where theinfimum is taken over all measures m ∈ P(TM) which are invariantunder the Euler-Lagrange flow (generated by L).1445. (Fathi [20]) A continuous function u : M → R is a viscosity solutionof H(x,∇u(x)) = −c if and only if it is Lipschitz and u is a negativeweak KAM solution (i.e. S−t u− ct = u). In particular, the statementis false if c is replaced with any other constant.In the language of transfers, the cost-minimizing transport is both aforward and backward linear transfer, with forward (resp. backward) Kan-torovich operators given by T+t f(x) = Vf (t, x) and T−t g(y) = Wtg(0, y),whereVf (t′, x) = inf{f(γ(0)) +∫ t′0L(γ(s), γ˙(s))ds ; γ ∈ C1([0, t′),M), γ(t′) = x}and W tg(t′, y) the value functionalW tg(t′, y) = sup{g(γ(t′))−∫ tt′L(γ(s), γ˙(s))ds ; γ ∈ C1([0, t′),M), γ(0) = x}.Observe that Vf (t, x) = S−f(x), while W tg(0, y) = S+g(y). Hence (withunfortunate signs), T+t f = S−t f(x), while T−t f(x) = S+t f(x). Note also thetranslation of terminology in this setting: Our backward weak KAM solu-tions are Fathi’s positive weak KAM solutions, while the analogous forwardweak KAM solutions are Fathi’s negative weak KAM solutions.As mentioned in Section 5.4.1 above, the negative (resp. positive) weakKAM solutions are the image of the Kantorovich operators T+∞ (resp. T−∞),and are given byT−∞f(x) = sup{f(y)−c∞(x, y) ; y ∈M} and T+∞f(y) = inf{f(x)+c∞(x, y) ; x ∈M}where c∞(x, y) := lim inft→∞(ct(x, y)−ct), with c = limt→∞ infµ,ν∈P(M) Tt(µ,ν)t .The cost-minimizing transport Tc∞ with cost c∞, is then the idempotentbackward (and forward) linear transfer associated to T−∞ (or T+∞) which byduality we can write asinf{∫M×Mc∞(x, y)dpi(x, y) ; pi ∈ K(µ, ν)} = sup{∫MT+∞gdν−∫MT−∞◦T+∞gdµ ; g ∈ C(M)}.145It can be checked this is exactly statement 2 in Theorem 5.4.8 above. Inparticular, Theorem 5.4.3 provides us with statements 1,2, and 3, in theabove theorem.5.4.3 The Schro¨dinger semigroupRecall the Schro¨dinger bridge of Example 2.8.6. Let M be a compact Rie-mannian manifold and fix some reference non-negative measure R on pathspace Ω = C([0,∞],M). Let (Xt)t be a random process on M whose lawis R, and denote by R0t the joint law of the initial position X0 and theposition Xt at time t, that is R0t = (X0, Xt)#R. Assume R is the re-versible Kolmogorov continuous Markov process associated with the gener-ator 12(∆ − ∇V · ∇) and the initial probability measure m = e−V (x)dx forsome function V .For probability measures µ and ν on M , defineTt(µ, ν) := inf{∫MH(rxt , pix)dµ(x) ; pi ∈ K(µ, ν), dpi(x, y) = dµ(x)dpix(y)}(5.39)where dR0t(x, y) = dm(x)drxt (y) is the disintegration of R0t with respect toits initial measure m.Proposition 5.4.9. The collection {Tt}t≥0 is a semigroup of backward lin-ear transfers with Kantorovich operators Ttf(x) := logStef (x) where (St)tis the semi-group associated to R; in particular,Tt(µ, ν) = sup{∫Mfdν −∫MlogStefdµ ; f ∈ C(M)}. (5.40)The corresponding idempotent backward linear transfer is T∞(µ, ν) = H(m, ν),and its effective Kantorovich map is T∞f(x) := logS∞ef , where S∞g :=∫gdm.Proof. It is easy to see that for each t, Tt is monotone, 1-Lipschitz andconvex, and also satisfies Tt(f + c) = Ttf + c for any constant c. It followsthat T ∗t,µ(f) =∫M Ttf dµ for each t by Proposition 2.4.4. The semigroupproperty then follows from the semigroup (St)t and the property that Tt ?146Ts is a backward linear transfer with Kantorovich operator Tt ◦ Tsf(x) =logStSsef (x) = logSs+tef (x) = Tt+sf(x) by Corollary 5.1.5.Now we remark that it is a standard property of the semigroup (St)ton a compact Riemannian manifold, that under suitable conditions on V ,Stef → S∞ef , uniformly on M , as t → ∞, for any f ∈ C(M). Thisimmediately implies by definition of Tt, that Ttf → T∞f uniformly as t→∞for any f ∈ C(M). We then deduce from the 1-Lipschitz property, thatTt ◦ T∞f(x) = T∞f(x). We conclude that T∞ is a Kantorovich operatorfrom Theorem 5.4.3. Finally we see that T∞(µ, ν) isT∞(µ, ν) := sup{∫fdν −∫T∞fdµ ; f ∈ C(M)}= sup{∫fdν − log∫efdm ; f ∈ C(M)}= H(m, ν).5.5 Regularization and ergodic properties ofnon-continuous linear transfersThe assumption of weak∗ continuity in the last section was important fortaking limits of the iterates T−n g + nc. When T is not necessarily weak∗-continuous on M(X) and may even have infinite values for some (µ, ν) ∈P(X)×P(X), we can no longer guarantee that taking limits of iterates willyield a proper function (i.e. one that is real-valued at least somewhere).Thus we turn to alternative strategies. The first and most natural one is toreduce the situation to the bounded and continuous case via a regularizationprocedure.5.5.1 RegularizationLemma 5.5.1 (Regularisation of a backward linear transfer). Let W1(µ, ν)be the cost minimising optimal transport associated to the cost dX(x, y). Fora given backward linear transfer T : P(X)×P(X)→ R∪ {+∞}, define for147 > 0,T(µ, ν) :=(1W1)? T ?(1W1)= inf{1W1(µ, σ1) + T (σ1, σ2) + 1W1(σ2, ν) ; σ1, σ2 ∈ P(X)}.Then, T has the following properties:1. T is a weak∗ continuous backward linear transfer.2. inf{T(µ, ν) ; µ, ν ∈ P(X)} = inf{T (µ, ν) ; µ, ν ∈ P(X)}.3. T(µ, ν) ≤ T (µ, ν) and T(µ, ν)↗ T (µ, ν) as ↘ 0.4. T Γ-converges to T as ↘ 0.5. If {δx ; x ∈ X} ⊂ D1(T ), and T− , T−, denote the backward Kan-torovich operators associated to T, T , respectively, then for any g ∈USC(X), T− g(x)↘ Tg(x) as ↘ 0 .Proof. First note that since dX is continuous, the linear transfer W1 is weak-∗ continuous on P(X) (see e.g., [48], Theorem 1.51, p.40).1. We know that for each fixed  > 0, T is a weak∗ lower semi-continuouslinear backward transfer. To prove that it is continuous, assume µn → µ andνn → ν. By the lower semi-continuity, we have lim infn T(µn, νn) ≥ T(µ, ν).On the other hand, from the fact that lim supn infσ1,σ2 ≤ infσ1,σ2 lim supn,we havelim supnT(µn, νn) ≤ infσ1,σ2∈P(X){lim supn1W1(µn, σ1) + T (σ1, σ2) + lim supn1W1(σ2, νn)}= infσ1,σ2∈P(X){1W1(µ, σ1) + T (σ1, σ2) + 1W1(σ2, ν)}= T(µ, ν),which shows that T(µn, νn)→ T(µ, ν) as n→∞.2. Observe from the definition of T, thatinfµ,ν{T(µ, ν)} = infσ,σ′,µ,ν{1W1(µ, σ) + T (σ, σ′) + 1W1(σ′, ν)}.148Since W1 ≥ 0, it follows that for fixed σ, σ′, the minimal value when min-imising over all µ, ν is to take µ = σ, and ν = σ′, in which case the transportcost vanishes W1(µ, σ) = 0 = W1(σ′, ν).3. The inequality T(µ, ν) ≤ T (µ, ν) holds by selecting σ1 = µ andσ2 = ν and noting that W1(σ, σ) = 0 for every σ ∈ P(X). The monotoneproperty of  7→ T(µ, ν) is immediate by definition. Let now σ1, σ2 realisethe infimumT(µ, ν) = 1W1(µ, σ1 ) + T (σ1 , σ2 ) +1W1(σ2 , ν). (5.41)By selecting a further subsequence if necessary, we may assume that σ1 → σ1and σ2 → σ2 as → 0. Suppose now lim inf→0 T(µ, ν) <∞. Then it mustbe the case that W1(µ, σ1 ) → 0 and W1(σ2 , ν) → 0 as  → 0. This impliesσ1 = µ and σ2 = ν since W1 is a metric on P(X). Then (5.41) and weak∗lower semi-continuity of T impliesT (µ, ν) ≥ lim inf→0T(µ, ν) ≥ lim inf→0T (σ1 , σ2 ) ≥ T (µ, ν).If lim inf→0 T(µ, ν) = ∞, then necessarily T (µ, ν) = +∞ since T (µ, ν) ≥T(µ, ν) for all . In either case, we deduce that lim→0 T(µ, ν) = T (µ, ν).4. First recall that for Γ-convergence, one needs to prove(i) the Γ-lim inf inequality: For every sequence (µ, ν)→ (µ, ν), it holdsthat lim inf→0 T(µ, ν) ≥ T (µ, ν), and(ii) the Γ-lim sup inequality: There exists a sequence (µ, ν) → (µ, ν)such that lim sup→0 T(µ, ν) ≤ T (µ, ν).The Γ-lim sup inequality is immediate: Take (µ, ν) = (µ, ν), and theinequality follows from T ≤ T . For the Γ-lim inf inequality, we have bymonotonicity that T(µ, ν) ≥ T′(µ, ν) for  ≤ ′. The weak∗ lower semi-continuity of T′ therefore implieslim inf→0T(µ, ν) ≥ lim inf→0T′(µ, ν) ≥ T′(µ, ν).By 3) and letting ′ → 0, we obtain lim inf→0 T(µ, ν) ≥ T (µ, ν).5. First note that the monotonicity of T− g(x) is immediate from the149expressionT− g(x) = supσ∈P(X){∫Xgdσ − T(δx, σ)},together with the monotonicity of T. Fromlim sup→0supσ∈P(X){∫Xgdσ − T(δx, σ)}≥ supσ∈P(X){∫Xgdσ − lim inf→0T(δx, σ)}= supσ∈P(X){∫Xgdσ − T (δx, σ)}we immediately have lim inf→0 T− g(x) ≥ T−g(x). On the other hand, letj be a sequence such that T−j g(x)→ lim sup→0 T− g(x). Then there is σjsuch thatT−j g(x) = supσ∈P(X){∫Xgdσ − T(δx, σ)}=∫Xgdσj − Tj (δx, σj ).By selecting a further subsequence if necessary, we may assume σj → σ∗.Then we obtain with j →∞,lim sup→0T− g(x) = limj→∞T−j g(x) ≤∫Xgdσ∗ − lim infj→∞Tj (δx, σj )≤∫Xgdσ∗ − T (δx, σ∗)≤ supσ∈P(X){∫Xgdσ − T (δx, σ)}= T−g(x),where the second inequality was obtained from the Γ-convergence.Lemma 5.5.2. Let T be a backward linear transfer such that D1(T ) containsall the Dirac measures. Assume c(T ) := infµ T (µ, µ) < +∞ . and let Tbe the regularisation of T according to Lemma 5.5.1. Then, the followingproperties hold:1. c(T) := infµ,µ T(µ, µ) = limn→∞ infµ,ν∈P(X) T(µ,ν)n is the unique con-stant such that |(T)n(µ, ν)− nc(T)| ≤ C, for all n and all µ, ν.2. c(T)↗ c(T ) as ↘ 0.150Proof. Use Lemma 5.5.1 to regularise T to T. Then item 1 is simplyLemma 5.4.1 for T. For 2), note that since T(µ, µ) ≤ T (µ, µ), thenlim sup→0 c(T) ≤ c(T ). On the other hand, letc(T) = inf{T(µ, µ);µ ∈ P(X)} = T(µ, µ)for some µ. If µ¯ is a cluster point for (µ) as  → 0, the Γ-convergence ofT to T implieslim inf→0c(T) = lim inf→0T(µ, µ) ≥ T (µ¯, µ¯) ≥ c(T )which concludes the proof.5.5.2 Ergodic properties for non-continuous linear transfersvia regularisationWe now present some results for backward linear transfers which are onlyweak∗ lower semi-continuous.Theorem 5.5.3 (Ergodic properties for non-continuous linear transfers).Let T : P(X)×P(X)→ R ∪ {+∞} be a backward linear transfer such thatD1(T ) contains all the Dirac measures. Assume c(T ) := infµ∈P(X) T (µ, µ) <+∞, and that for some ′ > 0 (and thus for all 0 <  ≤ ′), we havec(T) = c(T ), (5.42)where T is the regularisation of T according to Lemma 5.5.1.Then, there exists an idempotent backward Kantorovich operator T−∞ :C(X) → USC(X) such that T− ◦ T−∞g + c(T ) = T−∞g for all g ∈ C(X).Consequently, its associated backward linear transfer T∞ is idempotent andtherefore A-factorisable.Proof. Consider the regularisation T of T . By Theorem 5.4.3, there exists aKantorovich operator T−∞, : C(X)→ C(X), such that T− ◦ T−∞,g+ c(T) =T−∞,g for all g ∈ C(X), and an idempotent transfer T∞,. In fact, fromTheorem 5.4.3 we may take T−∞,g = limn→∞[(T− )n ◦ T¯− g + nc(T)], where151T¯− g(x) := lim supn[(T− )ng(x)+nc(T)]. Under the assumption that c(T) =c(T ), the monotonicity in  for T− (see Lemma 5.5.1) actually extends tomonotonicity for T−∞,, in the following sense,T−∞,g ≤ T−∞,′g for all g, whenever  < ′.Define nowT−∞g(x) := inf>0T−∞,g(x) = lim→0T−∞,g(x).As an infimum of continuous functions, T−∞g is upper semi-continuous, butwe have to check that it is proper (i.e. not identically −∞).To see this, recall the monotonicity T′(µ, µ) ≤ T(µ, µ) ≤ T (µ, µ) for ≤ ′, for all µ. When ′ is small enough, the hypothesis implies c(T′) =c(T) = c(T ).Let now µ¯ achieve c(T) = c(T ). From the monotonicity,c(T ) = c(T′) ≤ T′(µ¯, µ¯)≤ T(µ¯, µ¯) = c(T) = c(T ),hence T′(µ¯, µ¯) = c(T ). By Theorem 5.4.3, we have T∞,′(µ¯, µ¯) = 0,which implies ∫Xgdµ¯ ≤∫XT−∞,′gdµ¯. (5.43)Extract a subsequence j of the µ¯ so that µ¯j → µ¯. We know by Γ-convergence that T (µ¯, µ¯) = c(T ), and (5.43) implies that∫Xgdµ¯ ≤∫XT−∞,′gdµ¯.Now let ′ → 0 to obtain by monotone convergence∫Xgdµ¯ ≤∫XT−∞gdµ¯. (5.44)In particular, we deduce that T−∞g is finite for µ¯-a.e. x, so T−∞g ∈ USC(X)152for every g ∈ C(X).Regarding the claim that T−∞ is a backward Kantorovich operator, wehave from Theorem 5.4.3 that T−∞, is a backward Kantorovich operator forevery  > 0. Therefore, the monotonicity, convexity, and affine on con-stants, properties hold in the monotone limit as  → 0. For the lowersemi-continuity, we can return to the proof of Theorem 5.4.3 and write anequivalent inequality as was given in 5.33,T−∞,g(x) ≤ T−∞,gk(x) + ‖g − gk‖∞.Now let → 0 to obtainT−∞g(x) ≤ T−∞gk(x) + ‖g − gk‖∞which gives the desired lower semi-continuity property for T−∞ and showsthat T−∞ is a backward Kantorovich operator.Regarding the idempotent property for T−∞, we can exploit the mono-tonicity and idempotent properties for the Kantorovich operator T−∞,. ThisyieldsT−∞,g = T−∞, ◦ T−∞,g ≥ T−∞ ◦ T−∞gso that with → 0 we obtain T−∞g ≥ T−∞ ◦ T−∞g. For the reverse inequality,we again use montonicity in  to have for  ≤ ′,T−∞,g = T−∞, ◦ T−∞,g ≤ T−∞,′ ◦ T−∞,gWe let  → 0 and use the monotone convergence Lemma 5.1.3 applied toT∞′ , and then let ′ → 0 to achieveT−∞g ≤ T−∞ ◦ T−∞gso that T−∞g = T−∞ ◦ T−∞g.DefineT∞(µ, ν) := sup{∫Xgdν −∫XT−∞gdµ ; g ∈ C(X)}153as the induced linear transfer from T−∞. Note that T∞(µ¯, µ¯) = 0 by (5.44), soT∞ is proper, and µ¯ ∈ A. In addition, T∞ is idempotent. By the monotoneconvergence Lemma 5.1.3, and the monotonicity T−h ≤ T− h, we haveT− ◦ T−∞g(x) + c(T ) = lim→0T− ◦ T−∞,g(x) + c(T )≤ lim→0T− ◦ T−∞,g(x) + c(T) = lim→0T−∞,g(x) = T−∞f(x).On the other hand, the monotonicity again T− ≤ T−′ for  < ′ givesT−∞,g = T− ◦ T−∞,g + c(T) ≤ T−′ ◦ T−∞,g + c(T )By Lemma 5.1.3 applied to T−′ and the sequence T−∞,g, we can pass thelimit in  through T ′− to obtainT−∞g(x) = lim→0T−∞,g(x) ≤ lim→0T−′ ◦ T−∞,g(x) + c(T ) = T−′ ◦ T−∞g(x) + c(T ).Now we let ′ → 0 and use Property 5 of Lemma 5.5.1 to obtain T−∞g(x) ≤T− ◦ T−∞g(x) + c(T ), and thus obtaining equality.The hypothesis c(T) = c(T ) in Theorem 5.5.3 seems possibly difficultto check. In the next proposition we give more easily checkable sufficientconditions (see in particular item 1 below) for when this equality holds.Proposition 5.5.4. Let T be a backward linear transfer on P(X)×P(X).In any of the following cases,1. inf{T (µ, ν);µ, ν ∈ P(X)} = inf{T (µ, µ) ; µ ∈ P(X)},2. If T is both a backward and forward linear transfer, is symmetric, andthe operator T− associated to T maps every continuous function to aL-Lipschitz function (L > 0),we have c(T) = c(T ) for all  > 0 small enough; in particular, the equalityholds for all  > 0 in the first case, and for all 1L >  > 0 in the second.154Proof. 1) Note that property 2 of Lemma 5.5.1 saysinf{T (µ, ν);µ, ν ∈ P(X)} = inf{T(µ, ν);µ, ν ∈ P(X)}.At the same time, since c(T) ≤ c(T ) (see property 3 of Lemma 5.5.2),together with the assumption inf{T (µ, ν);µ, ν ∈ P(X)} = inf{T (µ, µ) ; µ ∈P(X)}, we getinfµ∈P(X)T(µ, µ) = c(T) ≤ c(T ) = infµ∈P(X)T (µ, µ) = infµ,ν∈P(X)T (µ, ν)= infµ,ν∈P(X)T(µ, ν)≤ infµ∈P(X)T(µ, µ)so all the inequalities are in fact equalities, and therefore c(T) = c(T ).2) WriteT(µ, µ) = infσ1,σ2{1W1(µ, σ1) + T (σ1, σ2) + 1W1(σ2, µ)}≥ infσ1,σ2{1W1(µ, σ1)− T (σ1, σ1) + T (σ1, σ2) + 1W1(σ2, µ)}+ infσ1{T (σ1, σ1)}≥ infσ1,σ2,σ3{1W1(µ, σ1)− T (σ1, σ3) + T (σ3, σ2) + 1W1(σ2, µ)}+ infσ1{T (σ1, σ1)}= (1W1) ? (−T ) ? T ? (1W1)(µ, µ) + c(T ).It suffices to show that infµ(1W1) ? (−T ) ? T ? (1W1)(µ, µ) ≥ 0, since thatwould imply based on the above inequalities, that c(T) ≥ c(T ) (the reverseinequality we already have by Lemma 5.5.2). To this end, first we can write(−T ) ? T (µ, ν) = infσ∈P(X){−T (µ, σ) + T (σ, ν)}= infg∈C(X)infσ∈P(X){∫XT−gdµ−∫Xgdσ + T (σ, ν)}= infg∈C(X){∫XT−gdµ+∫XT+(−g)dν},where we have used infσ∈P(X){∫X(−g)dσ + T (σ, ν)} =∫X T+(−g)dν since155T is a forward linear transfer. Then with the notation that S− g(x) :=supy∈X{g(y) − 1dX(x, y)} is the backward Kantorovich operator for 1W ,we arrive at(1W ) ? (−T ) ? T ?(1W )(µ, µ)= infσ1,σ2{1W (µ, σ1) + (−T ) ? T (σ1, σ2) + 1W (σ2, µ)}= infginfσ1,σ2{1W (µ, σ1) +∫XT−gdσ1 +∫XT+(−g)dσ2 + 1W (σ2, µ)}= infg{−∫XS− (−T−g)dµ−∫XS− (−T+(−g))dµ}= infg{−∫XS− (−T−g)dµ−∫XS− (T−g)dµ} (5.45)where we have used the fact that T−g = −T+(−g) since T is symmetric.Finally, we make use of the following property: if h ∈ C(X) is L-Lipschitzfor L < 1 , then S− (h) = h. Indeed, to see this, writeS− h(x) = supy∈X{h(y)− 1dX(x, y)}= supy∈X{h(y)− h(x)− 1dX(x, y)}+ h(x)≤ supy∈X{(L− 1)dX(x, y)}+ h(x) = h(x)the reverse inequality following similarly. Hence continuing from 5.45, wehave(1W ) ? (−T ) ? T ? (1W )(µ, µ) = infg{−∫X(−T−g)dµ−∫XT−gdµ}= 0,1565.5.3 ExamplesThe assumption inf{T (µ, ν);µ, ν ∈ P(X)} = inf{T (µ, µ) ; µ ∈ P(X)} issatisfied by a number of examples of which we present a few in this section.Example 5.5.5. Let T be the backward linear transfer associated to theconvex energy I(ν), that is T (µ, ν) := I(ν) and T−g(x) = I∗(g) (see Sec-tion 2.5.1). It is immediate that infµ,ν T (µ, ν) = infµ T (µ, µ) since T onlydepends on ν, so T satisfies the conditions of Theorem 5.5.3.We have c(T ) = infµ T (µ, µ) = infν I(ν). The operator T−∞g(x) :=I∗(g) + c(T ), where I∗ is the Legendre transform of I, is an idempotentbackward linear transfer that maps into the backward weak KAM solutionsfor T . Indeed, we can verifyT− ◦ T−∞g(x) + c(T ) = I∗(T−∞g) + c(T )= supν∈P(X){∫XT−∞gdν − I(ν)}+ c(T )= T−∞g − infν∈P(X)I(ν) + c(T ) (since T−∞g is a constant in x)= T−∞g.We can also observe that T−∞ is idempotent since T−∞ ◦ T−∞g = I∗(T−∞g) +c(T ) = T−∞g exactly as the computation above.The associated idempotent backward linear transfer is T∞(µ, ν) = I(ν)−c(T ), with A = {σ ; I(σ) = infν I(ν)}.Example 5.5.6. Consider the continuous point transformation of Example2.5.7, i.e.,T (µ, ν) =0 if ν = F#µ,+∞ otherwise,with T−g(x) = g(F (x)) for a continuous map F : X → X. The Krylov–Bogolyubovtheorem (see e.g. [51], Theorem 1.1) then says that there exists a measureµ ∈ P(X) such that F#µ = µ; therefore the hypotheses for Theorem 5.5.3are readily checked to be satisfied.157Suppose for simplicity that X ⊂ R. We then claim thatT−∞g(x) = ĝ ◦ F∞(x)is an idempotent backward Kantorovich operator mapping into the set ofbackward weak KAM solutions, where ĝ ◦ F∞ is the upper semi-continuousenvelope of g ◦ F∞, andF∞(x) := lim supm→∞Fm(x)where Fm denotes composition of F m-times. Indeed we haveT− ◦ T−∞g(x) = ĝ ◦ F∞(F (x))= inf{h(F (x)) ; h ∈ USC(X), h ≥ g ◦ F∞}= inf{h(x) ; h ∈ USC(X), h ≥ g ◦ F∞}= ĝ ◦ F∞(x) = T−∞g(x)where we have used the fact that if h ∈ USC(X) with h ≥ g ◦ F∞, thenh◦F ∈ USC(X) with h◦F ≥ g◦F∞ (here we use the fact that F∞◦F = F∞).The upper semi-continuous envelope is in general necessary; consider forexample X = [0, 1] and F (x) = x2, theng 7→ g ◦ F∞(x) =g(0) if x ∈ [0, 1)g(1) if x = 1,although idempotent and satisfying T− ◦ (g ◦ F∞) = g ◦ F∞, does not ingeneral belong to USC(X), unless g(0) ≤ g(1).Another idempotent operator mapping into backward weak KAM solu-tions when Fm(x) has more than one subsequential limit, isg 7→ g(lim infm→∞ Fm(x)) =: g(F∞(x)).For example, on X = [0, 1] if F (x) = 1 − x, then F∞(x) = max{x, 1 − x},while F∞(x) = min{x, 1− x}.158Example 5.5.7. Recall Example 2.5.7: With X = Y , for any lower semi-continuous A : X → R, consider T−g(x) = g(x)−A(x) and the correspond-ingT (µ, ν) =∫X Adµ if ν = µ and∫X Adµ < +∞,+∞ otherwise.We have infµ,ν∈P(X) T (µ, ν) = infµ∈P(X) T (µ, µ), so T satisfies the hypothe-ses of Theorem 5.5.3. Thenc(T ) = infµ∈P(X)T (µ, µ) = infx∈XA(x)andT−∞g(x) =g(x) if A(x) = c(T )−∞ otherwise.We know that T−∞g is proper since A achieves its infimum. To see thatT−∞g ∈ USC(X), note that if xk → x, then T−∞g(xk) is either −∞ or g(xk).In either case, we have lim supk→∞ T−∞g(xk) ≤ g(x) by continuity of g. Forthe lower semi-continuity property of T−∞, suppose gk → g in C(X) for thesup norm. If A(x) = c(T ), then lim infk→∞ T−∞gk(x) = lim infk→∞ gk(x) =g(x) = T−∞g(x); otherwise both T−∞gk(x) and T−∞g(x) are −∞; in either caselim infk→∞ T−∞gk(x) ≥ T−∞g(x).We can check the backward weak KAM solution,T− ◦ T−∞g(x) + c(T ) = T−∞g(x)−A(x) + c(T )=g(x)−A(x) + c(T ) if A(x) = c(T )−∞ otherwise.=g(x) if A(x) = c(T )−∞ otherwise.= T−∞g(x).159For the idempotent property, note thatT−∞ ◦ T−∞g(x) =T−∞g(x) if A(x) = c(T )−∞ otherwise. =g(x) if A(x) = c(T )−∞ otherwise. = T−∞g(x).The corresponding backward linear transfer T∞ isT∞(µ, ν) =0 if µ = ν is supported on the set {x ∈ X ; A(x) = c(T )}+∞ otherwiseand A = {µ ∈ P(X) ; spt(µ) ⊂ {x ; A(x) = c(T )}}.Note that if T−g(x) = g ◦ F (x) − A(x) for a continuous A, then theresulting linear transferT (µ, ν) =∫X Adµ if ν = F#µ,+∞ otherwise,in general fails the assumption inf(µ,ν) T (µ, ν) = infµ T (µ, µ). The latterminimisation, infµ T (µ, µ), is of interest in ergodic optimisation for expand-ing dynamical systems. We discuss more on this problem in Section 5.6.1.Example 5.5.8. Recall the Skorokhod transfer of Example 2.5.8 but nowwe let the stopping time τ be deterministic, and define, for measures µ, ν,on a compact Riemannian manifold X, the linear transfer,T (µ, ν) =0 if B0 ∼ µ and B1 ∼ ν+∞ otherwise.Then T−g(x) = E [g(B1)|B0 = x] = P1g(x), where {Pt}t>0 is the heat semi-group. Note that infµ,ν T (µ, ν) = infµ T (µ, µ) < +∞ since the volumemeasure λX (i.e., the uniform probability measure on X), is invariant, soT (λX , λX) = 0 (and in particular c(T ) = 0). Therefore the conditions ofTheorem 5.5.3 are satisfied.A corresponding idempotent backward Kantorovich operator is given by160T−∞g(x) =∫X gdλX . Indeed, we can check thatT− ◦ T−∞g(x) + c(T ) = P1(T−∞g) = T−∞gsince T−∞g is constant.The idempotent backward linear transfer is given byT∞(µ, ν) =0 if ν = λX+∞ otherwiseand A = {µ ; T∞(µ, µ) = 0} = {λX}.5.5.4 Interpolation and ergodic propertiesWe saw in the previous sections that the condition infµ,ν T (µ, ν) = infµ T (µ, µ) <+∞ is sufficient to apply Theorem 5.5.3. In this section, we apply a trans-formation to T to achieve another backward linear transfer T˜ , that has theproperty, infµ,ν T˜ (µ, ν) = infµ T˜ (µ, µ) < +∞.Lemma 5.5.9. Let S : C(X)→ C(X) be a Markov operator (i.e., a boundedlinear positive operator such that S1 = 1) and let S∗ :M(X)→M(X) be itsadjoint. Given a backward linear transfer T on P(X)×P(X) and λ ∈ (0, 1),defineT˜ (µ, ν) := T (µ, λS∗µ+ (1− λ)ν).Then, T˜ is a backward linear transfer with Kantorovich operatorT˜−g(x) := T−(11− λg)(x)− λ1− λSg(x)Proof. We first note that ν 7→ T˜ (µ, ν) is convex, weak∗ lower semi-continuous,and bounded below. In particular, to verify convexity, we write for β ∈ [0, 1]161and ν1, ν2 ∈ P(X),T˜ (µ, (1− β)ν1 + βν2) = T (µ, λS∗µ+ (1− λ)((1− β)ν1 + βν2))= T (µ, (1− β)[λS∗µ+ (1− λ)ν1] + β[λS∗µ+ (1− λ)ν2])≤ (1− β)T (µ, λS∗µ+ (1− λ)ν1) + βT (µ, λS∗µ+ (1− λ)ν2).Now we compute,(T˜µ)∗(g) = supσ∈P(X){∫Xgdσ − T˜ (µ, σ)}= supσ∈P(X){∫Xgdσ − T (µ, λS∗µ+ (1− λ)σ)}.Let now ν := λS∗µ+ (1− λ)σ; we obtain σ = 11−λν − λ1−λS∗µ. Hence aftersubstitution we obtain(T˜µ)∗(g) = supν∈P(X){∫X11− λgdν − T (µ, ν)} −λ1− λ∫YSgdµ=∫X[T−(11− λg)− λ1− λSg]dµ.Theorem 5.5.10. Let T be a backward linear transfer on P(X) × P(X)where X ⊂ Rn is compact, and suppose (µ0, ν0) is such that T (µ0, ν0) =inf(µ,ν) T (µ, ν) and µ0 has density with respect to Lebesgue. Then, for everyλ ∈ (0, 1), there exists a convex function φλ such that the backward lineartransfer given byT˜ (µ, ν) := T (µ, λ(∇φλ)#µ+ (1− λ)ν) (5.46)is such thatinfµ,ν∈P(X)T˜ (µ, ν) = infµ∈P(X)T˜ (µ, µ).In particular, the conclusion of Theorem 5.5.3 holds for T˜ , and there exists162a backward Kantorvich operator T−∞ : C(X)→ USC(X) such thatT− ◦ T−∞g + T (µ0, ν0) = λ (T−∞g) ◦ ∇φλ + (1− λ)T−∞g for all g ∈ C(X).Proof. The fact that T˜ is a backward linear transfer follows by Lemma 5.5.9.By assumption there is µ0, ν0 ∈ P(X) such that T (µ0, ν0) = infµ,ν T (µ, ν)and µ0 has density with respect to Lebesgue. Therefore, by Brenier’s the-orem (see e.g. [59], Corollary 2.30) there exists a convex function φλ suchthat ∇φλ#µ0 = (1− 1λ)µ0 + 1λν0.Consider now the backward linear transferT˜ (µ, ν) := T (µ, λ∇φλ#µ+ (1− λ)ν).Since λ∇φλ#µ0 + (1 − λ)µ0 = λ(1 − 1λ)µ0 + ν0 + (1 − λ)µ0 = ν0, we haveT˜ (µ0, µ0) = T (µ0, ν0) < +∞, and thereforeinfµT˜ (µ, µ) ≥ infµ,νT˜ (µ, ν) ≥ infµ,νT (µ, ν) = T (µ0, ν0) = T˜ (µ0, µ0) ≥ infµT˜ (µ, µ),hence infµ,ν T˜ (µ, ν) = infµ T˜ (µ, µ) = T (µ0, ν0).. The backward linear trans-fer T˜ therefore satisfies the hypotheses of Theorem 5.5.3, so there exists anidempotent backward Kantorovich operator T˜−∞ suchT˜− ◦ T˜−∞g + c(T˜ ) = T˜−∞g.By Lemma 5.5.9, T˜−g = T−( 11−λg)− λ1−λg ◦ ∇φλ, so thatT−(11− λT˜−∞g)−λ1− λ(T˜−∞g) ◦ ∇φλ + c(T˜ ) = T˜−∞g.In other words, by setting T−∞g :=11−λ T˜−∞g, we haveT− ◦ T−∞g + c(T˜ ) = λ (T−∞g) ◦ ∇φλ + (1− λ)T−∞g.1635.6 Ergodic properties for non-continuous lineartransfers with control on convergence to c(T )We saw in Section 5.5, that under the hypothesis c(T) = c(T ) for smallenough  > 0, we obtained certain ergodic properties described in Theorem5.5.3. We are interested in this section in obtaining results for backwardlinear transfers that do not satisfy this hypothesis. In particular, we shallfocus on the existence of functions h, possibly in USCσ(X), that satisfyT−h+ c(T ) = h (we shall not be concerned with trying to find a backwardKantorovich operator T−∞ as in the previous sections).A preliminary idea for constructing such a function h is very simple:For a backward Kantorovich operator T−, consider ϕ := lim supn→∞(T−n g+nc(T )). Then supm≥n(T−n g+ nc(T )) is a monotone decreasing sequence, soT−ϕ+ c(T ) ≥ ϕ, and thus T−n ϕ+nc(T ) is a monotone increasing sequence,so we can define h := limn→∞(T−n ϕ+nc(T )), and we have T−h+ c(T ) = h.One can also sketch a similar construction using the liminf instead of thelimsup.The issue with this construction concerns two things: (i) lim supn→∞(T−n g+nc(T )) (or lim infn→∞(T−n g+nc(T ))) may not be proper, and (ii) these func-tions are usually not in a “good enough space” (usually only in USCσ(X))to pass the monotone limit through T− (see Lemma 5.1.3). This sectiontherefore is concerned with finding some additional assumptions for whichto deal with (i) and (ii).We begin with a technical lemma concerning mainly with some prelimi-nary estimates which will help us to deal with issue (i) later in this section.Recall that the constant c(T ) := infµ T (µ, µ) = supn∈N inf(µ,ν) Tn(µ,ν)n < +∞.Lemma 5.6.1. Let T : P(X) × P(X) → R ∪ {+∞} be a backward lineartransfer such that D1(T ) contains the Dirac measures. Assume that c(T ) =infµ∈P(X) T (µ, µ) < +∞. Then, the following properties hold:1641. For each g ∈ C(X) and µ ∈ P(X), we have−T (µ, µ) ≤ lim infn→∞1n∫XT−n g dµ≤ lim supn→∞1n∫XT−n gdµ ≤ −c(T ).Consequently if T (µ¯, µ¯) = c(T ), then for each g ∈ C(X),limn→∞1n∫XT−n g dµ¯ = c(T ).2. For every n ∈ N and g ∈ USCb(X), we havesupx∈X(T−n g(x) + nc(T )) ≥ infy∈Xg(y). (5.47)3. Suppose there exists K > 0 such thatlim supn→∞{nc(T )− infµ,ν∈P(X)Tn(µ, ν)} ≤ K. (5.48)Then for all g ∈ C(X),supx∈Xlim supn→∞(T−n g(x) + nc(T )) ≤ supy∈Xg(y) +K. (5.49)Proof. 1)Write for the upper bound,∫XT−n gdµ = sup{∫Xg dν − Tn(µ, ν) ; ν ∈ P(X)}≤ sup(g)− inf(µ,ν)Tn(µ, ν).Dividing by n and recalling that c(T ) = limn→∞ inf(µ,ν) Tn(µ,ν)n we deducethe stated upper bound for lim supn→∞1n∫X T−n gdµ. For the lower bound,165write ∫XT−n gdµ = sup{∫Xg dσ − Tn(µ, σ) ; σ ∈ P(X)}≥∫Xg dµ− Tn(µ, µ)≥∫Xg dµ− nT (µ, µ).2) Writesupx∈X(T−n g(x) + nc(T )) = supµ∈P(X)∫X(T−n g(x) + nc(T )) dµ= supµ∈P(X)supσ∈P(X){∫Xgdσ − Tn(µ, σ) + nc(T )}≥ infXg − infσ,µTn(µ, σ) + nc(T )≥ infXg.The latter inequality follows from infµ,σ Tn(µ, σ) ≤ nc(T ) (see Proposition5.2.1).3) Writesupx∈Xlim supn→∞(T−n g(x) + nc(T )) ≤ lim supnsupx∈XT−n g(x) + nc(T )= lim supnsupx∈Xsupσ∈P(X){∫Xgdσ − Tn(δx, σ) + nc(T )}≤ sup(g) + lim supn{nc(T )− infx∈Xinfσ∈P(X)Tn(δx, σ)}≤ sup(g) + lim supn{nc(T )− inf(µ,ν)Tn(µ, ν)}≤ sup(g) +K.The following lemma is perhaps not surprising; the key point is that amonotone decreasing sequence is guaranteed to be proper.166Lemma 5.6.2. Suppose T is a backward linear transfer on P(X) × P(X)such that c(T ) = infµ∈P(X) T (µ, µ) < +∞ and supx∈X infν∈P(X) T (δx, ν) <+∞. If g ∈ USCb(X) is such that {T−n g + nc(T )}n∈N is a decreasing se-quence in n, then h(x) := limn→∞(T−n g(x)+nc(T )) belongs to USC(X) (inparticular, it is proper), and T−h+ c(T ) = h.Proof. The assumption supx∈X infν∈P(X) T (δx, ν) < +∞ implies that T−maps USCb(X) to USCb(X) (see Lemma 5.1.1). Therefore T−n g + nc(T ) isa decreasing sequence in USCb(X), hence converges to its infimum h, whichis therefore upper semi-continuous. To see that h is proper, suppose for everyx, it holds that T−n g(x)+nc(T ) decreases to −∞. Then for each x0, there isn0 ∈ N such that T−n0g(x0) + n0c(T ) ≤ infy∈X g(y)− 1. Since T−n0g + nc(T )is upper semi-continuous, the inequality for T−n0g(x0) +n0c(T ) must hold ina neighbourhood of x0. Since X is compact and T−n g+nc(T ) is decreasing,it follows that there is M ∈ N such that T−Mg(x) +Mc(T ) ≤ infy∈X g(y)− 1for all x ∈ X. But from Lemma 5.6.1, we also conclude thatsupx∈XT−Mg(x) +Mc(T ) ≥ infx∈X g(x)and the two inequalities together implyinfx∈Xg(x) ≤ infx∈Xg(x)− 1,an impossibility since infx∈X g(x) > −∞ as g ∈ USCb(X). Therefore weconclude that h is proper and therefore belongs to USC(X). Then by themonotonicity convergence Lemma 5.1.3, we conclude thatT−h+c(T ) = limn→∞T−(T−n g+nc(T ))+c(T ) = limn→∞(T−n+1g+(n+1)c(T )) = h.The following theorem is then concerned with finding conditions whichguarantee the existence of such a g ∈ USCb(X) as in Lemma 5.6.2.Theorem 5.6.3. Suppose T is a backward linear transfer on P(X)×P(X)such that D1(T ) contains all Dirac measures. Assume:1671. c(T ) := infµ T (µ, µ) < +∞,2. supx∈X infν∈P(X) T (δx, ν) < +∞,3. there exists K > 0 such thatlim supn→∞{nc(T )− infµ,ν∈P(X)Tn(µ, ν)} ≤ K,4. there exists g ∈ C(X), such that the functionx 7→ lim infn→∞ (T−n g(x) + nc(T ))belongs to USCb(X).Then there exists h ∈ USC(X) such that T−h+ c(T ) = h on X.Proof. Note that the second hypothesis implies that T−n g ∈ USCb(X) forall n ∈ N and all g ∈ USCb(X) (see Lemma 5.1.1).We distinguish two cases:Case 1: There is g ∈ C(X) so that ∀x ∈ X, there exists n ∈ N withT−n g(x) + nc(T ) < g(x).Since T−n g is in USCb(X), it follows that T−n g + nc(T ) < g on an openneighborhood Bx ⊂ X of x. Indeed, otherwise if there was a sequence xk →x with T−n g(xk)+nc(T ) ≥ g(xk), then T−n g(x)+nc(T ) ≥ lim supk T−n g(xk)+nc(T ) ≥ g(x), a contradiction.Since X is compact, there exists a finite number {x1, . . . , xk} ⊂ X suchthat {Bxj}1≤j≤k cover X. Set N := max{nx1 , . . . , nxk} and define ϕN (x) :=inf1≤n≤N (T−n g(x) + nc(T )). We have ϕN ∈ USCb(X), and moreover noteby construction that for any x ∈ X, ϕN (x) ≤ g(x). By the monotonicityproperty of T−, we haveT−ϕN+c(T ) ≤ T−g+c(T ) and T−ϕN+c(T ) ≤ inf2≤n≤N+1{T−n g+nc(T )}.168Therefore, combining the two we deduce thatT−ϕN + c(T ) ≤ inf1≤n≤N{T−n g + nc(T )} = ϕN .It follows that the sequence {T−n ϕN + nc(T )}n is decreasing and ϕN ∈USCb(X). Hence by Lemma 5.6.2, we conclude the existence of h ∈ USC(X),such that T−h+ c(T ) = h.Case 2: We now assume that for any g ∈ C(X), there exists x ∈ X suchthatT−n g(x) + nc(T ) ≥ g(x) for all n ∈ N. (5.50)By assumption, there exists g ∈ C(X) such that g˜ := lim infn→∞(T−n g +nc(T )) belongs to USCb(X). Then (5.50) implies there exists x ∈ X suchthat g˜(x) ≥ g(x) > −∞, and by Lemma 5.6.1, supx∈X g˜(x) ≤ supx∈X g(x)+K.Since infm≥n{T−mg +mc(T )} is increasing to g˜ and g˜ is bounded above,we have by Lemma 5.1.3, thatT−g˜ + c(T ) = limn→∞T−( infm≥n{T−mg +mc(T )}) + c(T )≤ lim infn→∞ (T−n+1g + (n+ 1)c(T ))= g˜.It follows that the sequence {T−n g˜ + nc(T )}n∈N ⊂ USCb(X) is decreasingand g˜ ∈ USCb(X). Consequently, we may apply Lemma 5.6.2 to concludethat there exists a function h ∈ USC(X) such that T−h+ c(T ) = h.The following provides a sufficient condition for lim supn→∞{nc(T ) −infµ,ν∈P(X) Tn(µ, ν)} ≤ K of the previous theorem.Proposition 5.6.4. Let T be a backward linear transfer on P(X)× P(X)such that c(T ) = infµ∈P(X) T (µ, µ) < +∞. If T is bounded above on P(X)×169P(X), thenTn(µ, ν)n→ c(T ) uniformly on P(X)× P(X). (5.51)In particular, we have lim supn→∞{nc(T ) − infµ,ν∈P(X) Tn(µ, ν)} ≤ K forsome constant K > 0.Proof. Note that { supµ,ν Tn(µ,ν)n }n∈N is a sub-additive sequence that satisfies,supµ,ν Tn(µ, ν)n≥ infµ,νT (µ, ν) > −∞hence converges to its infimum (see e.g. [9] Lemma 1.18). Sincec(T ) = limn→∞infµ,ν Tn(µ, ν)nandinfµ,ν Tn(µ, ν)n≤ supµ,ν Tn(µ, ν)nwe conclude thatc(T ) ≤ infn∈Nsupµ,ν Tn(µ, ν)n.Therefore, for all n ∈ N, it holds thatinfµ,νTn(µ, ν) ≤ nc(T ) ≤ supµ,νTn(µ, ν),so|Tn(µ, ν)− nc(T )| ≤ supµ,νTn(µ, ν)− infµ,νTn(µ, ν). (5.52)At the same time, we have for any µ, ν ∈ P(X) (writing infP×PTn for infµ,ν∈P(X) Tn(µ, ν)for brevity),infP×PTn−2 + 2 infP×P T ≤ Tn(µ, ν) ≤ 2 supP×P T + infP×P Tn−2,170from which follows thatsupP×PTn − infP×P Tn ≤ 2 supP×P T + infP×P Tn−2 − infP×P Tn−2 − 2 infP×P T= 2 supP×PT − 2 infP×PT=: K <∞. (5.53)Combining (5.52) and (5.53), we conclude that|Tn(µ, ν)− nc(T )| ≤ K for all µ, ν ∈ P(X) and all n ∈ N,which implies Tn(µ,ν)n → c(T ) uniformly on P(X)× P(X).Here is another situation where we can obtain weak KAM solutions.Proposition 5.6.5. Suppose T is a backward linear transfer on P(X) ×P(X) such that c(T ) = infµ∈P(X) T (µ, µ) < +∞, and supx∈X infν∈P(X) T (δx, ν) <+∞. If there exists u, v ∈ USCb(X) such thatT−n u+ nv = u for all n ∈ N, (5.54)then there exists h ∈ USC(X) such that T−h + c(T ) = h on X, where T−is the backward Kantorovich operator associated to T .Proof. By (5.54) and Lemma 5.6.1, we have −v(x) = lim supn T−n un ≤ −c(T ).Therefore,T−n u+ nc(T ) ≤ u for all n ∈ N.Applying T−m and using the linearity of T−m with respect to constants, wefind T−m+nu+ nc(T ) ≤ T−mu, and henceT−m+nu+ (m+ n)c(T ) ≤ T−mu+mc(T ).So n 7→ T−n u + nc(T ) is decreasing, and u ∈ USCb(X). Consequently byLemma 5.6.2, there exists h ∈ USC(X) such that T−h+ c(T ) = h.1715.6.1 Linear transfers and ergodic optimizationThis section was developed jointly with Dorian Martino [40]. We shall con-sider here linear transfers where the associated Kantorovich maps are affineoperators that is of the form T−f(x) = Tf(x)−A(x), where T is a Markovoperator and A is a given function (observable). In this section, we shall seethat the presence of A allows the theory of transfers to incorporate ergodicoptimization for expanding dynamical systems (see, for instance, [25]). Forsimplicity, we shall focus here on the case where the linear Markov operatoris given by a point transformation σ.We will in this section make use of the Fenchel-Rockafellar duality, andthus for convenience we record the statement of this theorem here.Theorem 5.6.6 (Fenchel-Rockafellar duality (see e.g. [59] Theorem 1.9)).Let E be a normed vector space, E∗ its topological dual space, and h1, h2 :E → R∪{+∞} two convex functions on E. If there exists z0 ∈ E such thath1(z0) < +∞, h2(z0) < +∞, and h1 is continuous at z0,theninfz∈E{h1(z) + h2(z)} = supz∗∈E∗{−h∗1(−z∗)− h2(z∗)}where h∗1, h∗2, is the Fenchel-Legendre transform of h1, h2, respectively,h∗1(z∗) := supz∈E{〈z∗, z〉 − h1(z)}.Proposition 5.6.7. Let σ : X → X be continuous and surjective, andassume there is a (sequentially) compact space Y such that for each y ∈ Y ,there exists a compact subset Xy of X, and a continuous map τy : Xy → X,such that σ ◦ τy(x) = x for all x ∈ Xy.Define Yx := {y ∈ Y ; y ∈ Xy}, and assume the following continuityproperties: If xk is a sequence in X (resp. yk a sequence in Y ) with yk ∈ Yxk ,then if xk → x and yk → y, we have y ∈ Yx, and τyk(xk)→ τy(x). Moreover,172assume that the maps τy are unique, in the sense that if τy(x) = τy′(x) forall x, then y′ = y.Let A ∈ C(Y×X) be a continuous function and consider the cost functionc : X ×X → R ∪ {+∞} defined byc(z, x) :=inf{A(y, x) ; y ∈ Yx, τy(x) = z} if σ(z) = x+∞ otherwise.Then, c is lower semi-continuous, and the optimal mass transport T asso-ciated to the cost c is simply the backward, and forward, linear transferT (µ, ν) =∫X A¯dµ if ν = σ#µ+∞ otherwise,which has backward (resp. forward) Kantorovich operator given byT−g(x) = g(σ(x))− A¯(x), (resp., T+f(x) = infy∈Yx{f(τy(x)) +A(y, x)}).where A¯(x) := c(x, σ(x)).Proof. To see that c is lower semi-continuous, suppose xk → x and zk → z.If for all but finitely many (zk, xk) we have c(zk, xk) = +∞, then there isnothing to prove. Therefore assume σ(zk) = xk, and hencec(zk, xk) = inf{A(y, xk) ; y ∈ Yxk , τy(xk) = zk}Since A is continuous and {y ∈ Yxk ; τy(xk) = zk} is a closed subset of thecompact set Yxk , the infimum is achieved by some yk, i.e.c(zk, xk) = A(yk, xk)The sequence {yk} ⊂ Y and Y is sequentially compact, so we may select aconvergent subsequence (that we relabel back to yk). This y has the property173that y ∈ Yx and τy(x) = z by assumption. Consequently,lim infk→∞c(zk, xk) = A(y, x) ≥ inf{A(y, x) ; y ∈ Yx, τy(x) = z} = c(z, x).We now compute the optimal transportT (µ, ν) = infpi∈K(µ,ν)∫X×Xc(z, x)dpi(z, x).If ν 6= σ#µ, then pi ∈ K(µ, ν) will give non-zero mass to a region {(z, x)}where σ(z) 6= x. Since c(z, x) = +∞ there, we deduce that T (µ, ν) = +∞in this case. Otherwise assume ν = σ#µ. Then pi ∈ K(µ, ν) is supported on{(z, σ(z)) ; z ∈ X}, soT (µ, ν) = infpi∈K(µ,ν){∫X×Xc(z, σ(z))dpi(z, x)}=∫Xc(z, σ(z))dµ=∫XA¯dµ.We know from Section 2.6 that the backward (resp. forward) Kantorovichoperator is given byT−g(z) = supx∈X{g(x)− c(z, x)} = g(σ(z))− c(z, σ(z)) = g ◦ σ(z)− A¯(z).andT+f(x) = infz∈X{f(z) + c(z, x)} = infy∈Yx{f(τy(x)) + c(τy(x), x)}.The final observation to make for T+ is thatc(τy(x), x) = inf{A(y′, x) ; y′ ∈ Yx τy′(x) = τy(x)}hence by assumption y′ = y, so that c(τy(x), x) = A(y, x).Theorem 5.6.8. In the set-up described in the previous Proposition 5.6.7,174let Pσ(X) denote the subset of probability measures in P(X) which are in-variant under σ. Let Xˆ denote the subset of Y × X consisting of points(y, x) such that x ∈ Xy, and define M0(Xˆ) as the subset of “holonomic”probability measures µˆ ∈ P(Xˆ) satisfying ∫Xˆ [f(τy(x))− f(x)]dµˆ = 0.Then, the following duality formulae holds:c(T ) := infµ∈P(X)T (µ, µ) = inf{∫XA¯(x) dµ(x);µ ∈ Pσ(X)}= inf{∫XˆA(y, x)dµˆ(y, x) ; µˆ ∈M0(Xˆ)}= supf∈C(X)infx∈X{f(x)− f(σ(x)) + A¯(x)}= supf∈C(X)infx∈Xinfy∈Yx{f(τy(x))− f(x) +A(y, x)}.Remark 5.6.9. The equality of the second and fourth lines has alreadybeen established in [26] for the setting of symbolic dynamics (see the nextsection).Proof. Let us first demonstrate the equalityinf{∫XA¯(x) dµ(x);µ ∈ Pσ(X)} = supf∈C(X)infx∈X{f(x)− f(σ(x)) + A¯(x)}= supf∈C(X)infx∈X{f(x)− T−f(x)}. (5.55)We can write using the duality of optimal transport,inf{∫XA¯(x) dµ(x);µ ∈ Pσ(X)} = infµ∈P(X)infpi∈K(µ,µ)∫X×Xc(x, y)dpi(x, y)= infµ∈P(X)supf∈C(X){∫X(f − T−f)dµ} (5.56)Comparing (5.55) and (5.56), it suffices to show thatsupf∈C(X)infx∈X{f(x)− T−f(x)} = infµ∈P(X)supf∈C(X){∫X(f − T−f)dµ}.We would therefore like to apply Sion’s minimax theorem (Theorem 2.7.6)175to the function F : P(X) × C(X) defined by F (µ, f) := ∫X(f − T−f)dµ,which is real-valued for all f ∈ C(X) and all µ ∈ P(X).First, µ 7→ F (µ, f) is weak∗ lower semi-continuous since f − T−f =f−f◦σ+A¯ is a lower semi-continuous function for each f ∈ C(X). Moreoverµ 7→ F (µ, f) is quasi-convex on P(X), i.e. {µ ∈ P(X) ; F (µ, f) ≤ λ} isconvex or empty for all λ ∈ R.On the other hand, f 7→ F (µ, f) is upper semi-continuous, since if fk → fin C(X), thenlim supk→∞∫X(fk − T−fk)dµ ≤∫X(f − lim infk→∞T−fk)dµ≤∫X(f − T−f)dµ.Moreover, f 7→ F (µ, f) is quasi-concave on C(X), i.e. {f ∈ C(X) ; F (µ, f) ≥λ} is convex or empty for all λ ∈ R.We may therefore apply Sion’s minimax theorem (Theorem (2.7.6)) toconclude thatinfµ∈P(X)supf∈C(X)∫X(f − T−f)dµ = supf∈C(X)infµ∈P(X)∫X(f − T−f)dµ= supf∈C(X)infx∈X∫X(f − T−f)dµ.Next let us demonstrate the equality,inf{∫XˆA(y, x)dµˆ(y, x) ; µˆ ∈M0(Xˆ)} = supf∈C(X)infx∈Xinfy∈Yx{f(τy(x))− f(x) +A(y, x)}As mentioned, this has already been established in [26] for the setting ofsymbolic dynamics (see Theorem 1 there). For this, we shall use Fenchel-Rockafellar duality. Recall Xˆ := {(y, x) ; x ∈ X, y ∈ Xy}, and define h1, h2 :C(Xˆ)→ R ∪ {+∞} be defined byh1(φ) := sup(y,x)∈Xˆ{−φ(y, x) +A(y, x)},176andh2(φ) =0 if φ is in the closure of {g ∈ C(Xˆ) ; g(y, x) = f(x)− f ◦ τy(x) for some f ∈ C(X)}+∞ otherwise.Note that h1(φ) − h1(φ˜) ≤ ‖φ − φ˜‖∞, so h1 is continuous on C(Xˆ). Tocompute their respective Legendre transforms, we haveh∗1(µˆ) = supφ∈C(Xˆ){∫Xˆφdµˆ− h1(φ)}.If µˆ(Xˆ) 6= −1, thenh∗1(µˆ) ≥ supλ∈R{λ(µˆ(Xˆ) + 1) + inf(y,x)∈XˆA(y, x)}and the supremum is +∞. Suppose now that µˆ(Xˆ) = −1, but −µˆ /∈ P(Xˆ).Then there exists a sequence of functions φn ∈ C(Xˆ) such that φn ≤ 0 andlimn→∞∫Xˆ φndµˆ = +∞. Then we haveh∗1(µˆ) ≥∫Xˆφndµˆ− h1(φn)≥∫Xˆφndµˆ− sup(y,x)∈XˆA(y, x)hence h∗1(µˆ) = +∞. Finally suppose −µˆ ∈ P(Xˆ). Thenh∗1(µˆ) ≥∫XˆAdµˆ− h1(A)=∫XˆAdµˆ,while also since φ + h1(φ) ≥ A, then∫Xˆ(φ + h1(φ))dµˆ ≤∫Xˆ Adµˆ (recall177µˆ(Xˆ) = −1), so we haveh∗1(µˆ) = supφ∈C(Xˆ){∫Xˆ(φ(y, x) + h1(φ))dµˆ}≤∫XˆAdµˆ.Therefore,h∗1(µˆ) =∫Xˆ Adµˆ if − µˆ ∈ P(Xˆ)+∞ otherwise.We also haveh∗2(µˆ) = supφ∈C(Xˆ){∫Xˆφdµˆ− h2(φ)}= supf∈C(X){∫Xˆ(f − f ◦ τy)dµˆ}.If∫Xˆ(f − f ◦ τy)dµˆ 6= 0 for some f , then substituting f with λf , λ ∈ R,implies this supremum will be +∞. Let S0 := {µˆ ∈ M(Xˆ) ;∫Xˆ(f ◦ τy(x)−f(x))dµˆ(y, x) = 0 for all f ∈ C(X)}. Then,h∗2(µˆ) =0 if µˆ ∈ S0+∞ otherwise.It now suffices to apply Fenchel-Rockafellar:infφ∈C(Xˆ){h1(φ) + h2(φ)} = supµˆ∈M(Xˆ){−h∗1(−µˆ)− h2(µˆ)}.Finally, to complete the string of equalities stated in the theorem, theequality c(T ) = infµ∈P(X) T (µ, µ) = inf{∫X A¯(x) dµ(x);µ ∈ Pσ(X)} is im-mediate from Proposition 5.6.7. In addition, we observe by definition of178A¯(x) = c(x, σ(x) that for any f ∈ C(X),infx∈X{f(x)− f(σ(x)) + A¯(x)} = infx∈Xinfy∈Yσ(x),τy(σ(x))=x{f(x)− f(σ(x)) +A(y, σ(x))}= infz∈Xinfy∈Yz{f(τy(z))− f(z) +A(y, z)},where the last equality holds by making the change of variable z := σ(x),along with the fact that σ is assumed to be surjective. It therefore followsthatsupf∈C(X)infx∈X{f(x)− f(σ(x)) + A¯(x)} = supf∈C(X)infx∈Xinfy∈Yx{f(τy(x))− f(x) +A(y, x)}which concludes the demonstration of the equalities stated in the theorem.Theorem 5.6.10. Suppose thatx 7→ lim infn→∞n−1∑k=0infµ∈Pσ(X)∫XA¯dµ− A¯(σk(x))belongs to C(X). Then there exists h ∈ USC(X) such thath(σ(x))− A¯(x) + c(T ) = h(x) for all x ∈ X, (5.57)equivalently,infy∈Yx{h(τy(x)) +A(y, x)} − c(T ) = h(x) for all x ∈ X. (5.58)That is, h satisfies T−h+ c(T ) = h, and T+h− c(T ) = h.Proof. To establish the existence of a function satisfying (5.57), note thatthis is equivalent to having a function h such that T−h(x) + c(T ) = h(x),hence it suffices to show that the assumptions of Theorem 5.6.3 are satisfied.For that, first note that by a theorem of Bogolyubov and Krylov ([51],Theorem 1.1), σ has an invariant measure, hence c(T ) = infµ T (µ, µ) < +∞.179In addition, we have for each x ∈ X,supx∈Xinfν∈P(X)T (δx, ν) ≤ supx∈XA¯(x) < +∞.In order to show the condition (5.48), we let for each n ∈ N, µn ∈P(X) be such that Tn(µn, (σn)]µn) = infµ,ν Tn(µ, ν). Select a subsequence(which we relabel back to n) so that νn := (σn)]µn converges to some µ¯ ∈P(X). Then the Ce´saro average 1n∑nk=1 νk converges to µ¯. Moreover, µ¯ isσ-invariant. Indeed,σ]µ¯ = limn→∞1nn∑k=1(σk)]µn = limn→∞(1nn−1∑k=0(σk)]µn +1n((σn)]µn − µn))= µ¯.Recall that A¯(x) = c(x, σ(x)) is lower semi-continuous, solim supn→∞∫XA¯dµ¯− 1nn−1∑k=0∫XA¯d((σk)]µn) ≤ 0.Therefore, by selecting a further subsequence (which again we may relabelback to n), we may assume that this subsequence has the property thatthere exists K > 0 such that∫XA¯dµ¯− 1nn−1∑k=0∫XA¯d((σk)]µn) ≤ Kn, n ∈ N.180We then obtain the desired estimate,lim supn→∞(nc(T )− infµ,ν∈P(X)Tn(µ, ν))≤ lim supn→∞(nT (µ¯, µ¯)− Tn(µn, (σn)]µn))= lim supn→∞(nT (µ¯, µ¯)−n−1∑k=0T ((σk)]µn, (σk+1)]µn))= lim supn→∞(n∫XA¯dµ¯−n−1∑k=0∫XA¯d((σk)]µn))≤ K.Finally note that for g ≡ 0,T−n (0) + nc(T ) =n−1∑k=0( infµ∈Pσ(X)∫XA¯dµ− A¯ ◦ σk).Since by assumption,x 7→ lim infn→∞n−1∑k=0( infµ∈Pσ(X)∫XA¯dµ− A¯ ◦ σk(x))belongs to C(X), it follows that lim infn→∞ T−n (0)+nc(T ) belongs to C(X).Hence all the hypotheses for application of Theorem 5.6.3 hold, and wededuce that there exists h ∈ USC(X) such thatT−h+ c(T ) = h.Remark 5.6.11. A conjecture is that the assumptionx 7→ lim infn→∞n−1∑k=0(infµ∈Pσ(X)∫XA¯dµ− A¯(σk(x)))belongs to C(X), holds when A is Ho¨lder continuous. This matches exactly181the condition on A required by Garibaldi and Lopes ([26], Theorem 4) - seethe next section.5.6.2 Ergodic optimization in the deterministic holonomicsettingWe now apply the result of the previous section to the setting of symbolicdynamics. Fix r ∈ N, and let M be an r×r transition matrix, whose entriesare either 0 or 1, specifying the allowable transitions. Denote byΣ = {x ∈ {1, ..., r}N ; ∀i ≥ 0, M(xi, xi+1) = 1}the set of admissible words, its dualΣ∗ = {y ∈ {1, ..., r}N ; ∀i ≥ 0, M(yi+1, yi) = 1},and consider the spaceΣˆ = {(y, x) ∈ Σ∗ × Σ ; M(y0, x0) = 1}.For each x ∈ Σ, we let Σ∗x = {y ∈ Σ∗ ; (y, x) ∈ Σˆ} and assume that∀x, Σ∗x 6= ∅. We will denote the words of Σ with their starting letters, i.e.,(x0, x1, ...) while the words in Σ∗ will be identified with their ending letters,i.e., (..., y1, y0). We consider Σ and Σ∗ as metric spaces with the distanced(x, x¯) = 2−min{j∈N ; xj 6=x¯j}. In particular, all these sets are compact.Consider now the two continuous maps σ : Σ→ Σ and τ : Σˆ→ Σ defined asσ(x0, x1, ...) = (x1, x2, ...) and τ(y, x) = (y0, x0, x1, ...). (5.59)The map σ can be considered as the time-evolution operator. We will denoteτ(y, x) by τy(x) and consider the set of holonomic probability measuresM0(Σˆ) :={µ ∈ P(Σˆ) ;∫Σˆf(τy(x))− f(x) dµ(y, x) = 0}.182E. Garibaldi and A. O. Lopes studied an Aubry-Mather theory for symbolicdynamics [26]; in particular, they prove the following results.Theorem 5.6.12 (Garibali and Lopes [26], Theorem 1, Theorem 4). Underthe set-up described above, given A ∈ C(Σˆ), define βA := maxµˆ∈M0(Σˆ)∫ΣˆA(y, x)dµˆ(y, x).ThenβA = inff∈C(Σ)max(y,x)∈Σˆ{A(y, x) + f(x)− f(τy(x))}.If A ∈ C0,θ(Σˆ) is θ-Ho¨lder continuous, then there exists a function u ∈C0,θ(Σ) such thatu(x) = minu∈Σ∗x{u(τy(x))−A(y, x) + βA}.By applying our results of the previous Section 5.6.1 in this symbolicdynamics setting, we obtain the following, using the notation βA of Theorem5.6.12.Proposition 5.6.13. Under the set-up described at the beginning of thissection, given A ∈ C(Σˆ), then the following hold:βA = supµ∈Pσ(Σ){∫ΣA¯dµ} (5.60)= inff∈C(Σ)supx∈Σ{f(σ(x))− f(x) + A¯(x)} (5.61)= inff∈C(Σ)sup(y,x)∈Σˆ{f(x)− f(τy(x)) +A(y, x)}, (5.62)where A¯(x) := sup{A(y, x) ; y ∈ Σ∗x, τy(σ(x)) = x}.Moreover, there exists h ∈ USC(Σ) such thatinfy∈Σ∗x{h(τy(x))−A(y, x) + β(A)} = h(x) ∀x ∈ Σ. (5.63)Proof. We will apply Proposition 5.6.7, Theorem 5.6.8, and Theorem 5.6.10,with −A instead of A, and with the following identifications:X := Σ, Y := Σ∗, Yx := Σ∗x, Xˆ := Σˆ,183and σ, τy, as in (5.59).5.6.3 Ergodic optimization in the stochastic holonomicsettingWe now propose the following model:Construct inductively, the following sequence: Let X0 ∈ Σ be a randomword, B0 ∈ Σ∗X0 a random “noise”, and let X¯0 := τB0(X0). We then letY 0 ∈ Σ∗¯X0be a random “control”, and consider X1 := τY 0(X¯0). We thenchoose B1 ∈ Σ∗X1 and let Y 1 ∈ Σ∗¯X1 .Iterating this process, an entire random past trajectory of X0 is repre-sented via the random family (Xn)n ∈ ΣN. The goal is to minimise the longtime average cost,limn→∞1nE[n−1∑i=0A(Y i, X¯i)]among all possible such choices.We assume that B0, Y 0, satisfy the following “martingale-type” prop-erty:E[f(Y 0, τB0(X0))|X0 = σ(x)] = E[f(Y 0, x)] for any f ∈ C(Σˆ). (5.64)For each such trajectory, consider for each n ∈ N, the measure µn ∈ P(Σˆ)defined via ∫Σˆφdµn :=1nn−1∑i=0E[φ(Y i, X¯i)], φ ∈ C(Σˆ).From (µn)n, one can extract a subsequence converging to some measureµ(Xi)i by compactness of P(Σˆ). We denoteM0 = {µ(Xi)i} ⊂ M(Σˆ),as the closure of all such µ(Xi)i . For f ∈ C(Σ) and (y, x) ∈ Σˆ, denote12Dyf(x) := f(τy(x))−f(x)−f(τy(x))− 2f(x) + f(σ(x))2=f(τy(x))− f(σ(x))2.184Note that the assumption made on the random noise B0 yields an Itoˆ-typeformula: For all f ∈ C(Σ), x ∈ Σ, with B0 ∈ Σ∗σ(x) Y 0 ∈ Σ∗τB0 (x),E[f(τY 0(τB0(σ(x))))− f(σ(x))] = E[DY0f(x)].Let alsoN0 = {µ ∈M(Σˆ) | ∀f ∈ C(Σ),∫ΣˆDyf(x)dµ(y, x) = 0},which is closed in M(Σˆ) as a kernel of a continuous linear map.Lemma 5.6.14. We have M0 ⊂ N0.Proof. Each measure µ(Xi)i ∈M0,∫ΣˆDyf(x)dµn(y, x) =1nE[n−1∑i=0f(Xi+1)− f(Xi)]=1nE[f(Xn)− f(X0)]which tends to zero as n→∞ since f is bounded.Theorem 5.6.15. With the above notation, we have the followinginfµ∈N0∩P(Σˆ)∫ΣˆAdµˆ = supf∈C(Σ)inf(y,x)∈ΣˆDyf(x) +A(y, x). (5.65)Ifx 7→ lim infn→∞(n infµˆ∈M0∩P(Σˆ)∫ΣˆAdµˆ− inf{n−1∑k=0E[A(Y k, τBk(Y¯ k)] ; Y k ∈ Σ∗τBk(Y¯ k)})belongs to C(Σ), where Y¯ k is defined recursively via Y¯ 0 = x, Y¯ k = τY k−1(τBk−1(Y¯k−1)),k ≥ 1, then there exists an g ∈ USC(Σ) such that h := −g satisfiesinfµ∈N0∩P(Σˆ)∫ΣˆAdµˆ = infy∈Σ∗x{Dyh(x) +A(y, x)}. (5.66)185Proof. The last equality in (5.65), i.e., is again an application of the Fenchel-Rockafellar duality, similar to the previous section. Indeed, consider thefunctions h1, h2 : C(Σˆ)→ R ∪ {+∞} defined byh1(φ) := sup(y,x)∈Σˆ{−φ(y, x) +A(y, x)}andh2(φ) :=0 if φ ∈ {g ∈ C(Σˆ) ; g(y, x) = Dyf(x) for some f ∈ C(Σ)}+∞ otherwise.Note that h1 and h2 are convex, with |h1(φ) − h1(φ˜)| ≤ ‖φ − φ˜‖∞. Tocompute their Legendre transform, we haveh∗1(µˆ) = supφ∈C(Σˆ){∫Σˆφdµˆ− h1(φ)}.We haveh∗1(µˆ) ≥ supλ∈R{(λ+ 1)µˆ(Σˆ) + inf(y,x)∈Σˆ{A(y, x)}and therefore if µˆ(Σˆ) 6= −1, the supremum is +∞. Suppose now thatµˆ(Σˆ) = −1, but −µˆ /∈ P(Σˆ). Then there exists a sequence of functionsφn ∈ C(Σˆ) such that φn ≤ 0 and limn→∞∫Σˆ φndµˆ = +∞. Thenh∗1(µˆ) ≥∫Σˆφndµˆ− h1(φn)≥∫Σˆφndµˆ− sup(y,x)∈ΣˆA(y, x)hence h∗1(µˆ) = +∞. Finally suppose −µˆ ∈ P(Σˆ). Then we haveh∗1(φ) ≥∫ΣˆAdµˆ− h1(A) =∫ΣˆAdµˆwhile on the other hand from φ + h1(φ) ≥ A, we have∫Σˆ(φ − h1(φ))dµˆ ≤186∫ΣˆAdµˆ (recall µˆ(Σˆ) = −1), so thath∗1(φ) = supφ∈C(Σˆ){∫Σˆ(φ− h1(φ))dµˆ}≤ supφ∈C(Σˆ){∫Σˆ(φ− φ−A)dµˆ} =∫ΣˆAdµˆ.Consequently,h∗1(µˆ) =∫ΣˆAdµˆ if µˆ ∈ P(Σˆ),+∞ otherwise.Similarly, we haveh∗2(µˆ) =0 if µˆ ∈ N0+∞ otherwise.Indeed, if µˆ /∈ N0, there exists f¯ ∈ C(Σ) such that∫ΣˆDyf¯(x)dµˆ(y, x) 6= 0.Hence, we replacing f¯ with λf¯ , λ ∈ R, we see thath∗2(µˆ) = supf∈C(Σ)∫ΣˆDyf(x)dµˆ(y, x) = +∞.If µˆ ∈ N0, by definition, h∗2(µˆ) = 0.Fenchel-Rockafellar then says thatinfφ∈C(Σˆ){h1(φ) + h2(φ)} = supµˆ∈M(Σˆ){−h∗1(−µˆ)− h∗2(µˆ)},which completes the proof of (5.65).We now provide the proof of (5.66): Consider the functional T : P(Σ)×P(Σ)→ R ∪ {+∞} defined viaT (µ, ν) := infE[A(Y 0, X¯0)]∣∣∣∣∣∣∣X0 ∼ νX¯0 = τB0(X0) B0 ∈ Σ∗X0X1 = τY 0(X¯0) ∼ µ Y 0 ∈ Σ∗¯X0187We claim that T is a forward linear coupling withT+f(x) := inf{E[f(τY 0(τB0(x)) +A(Y 0, τB0(x))] ; Y 0 ∈ Σ∗τB0 (x)}.Indeed, we have∫ΣT+fdν =∫Σ(inf{E[f(τY 0(τB0(x)) +A(Y 0, τB0(x))] ; Y 0 ∈ Σ∗τB0 (x)})dν= inf{E[f(τY 0(τB0(X0)) +A(Y 0, τB0(X0))] ; Y 0 ∈ Σ∗τB0 (X0), X0 ∼ ν}.Note then that supf∈C(Σ){∫Σ T+fdν−∫Σ fdµ} will be +∞, unless τY 0(τB0(X0)) ∼µ, in which case the terms in f cancel, leavingsupf∈C(Σ){∫ΣT+fdν −∫Σfdµ} = inf{E[A(Y 0, τB0(X0))] ; Y 0 ∈ Σ∗τB0 (X0), X0 ∼ ν, τY 0(τB0(X0)) ∼ µ}= T (µ, ν).We now show that the hypotheses for application of Theorem 5.6.3 tothe (backward) linear coupling T˜ (µ, ν) := T (ν, µ), are satisfied.First, it is easy to see that supx∈Σ infν∈P(Σ) T˜ (δx, ν) < +∞. Indeed,for a fixed x ∈ Σ, take any random noise B0 ∈ Σ∗x and random strategyY 0 ∈ Σ∗τB0 (x), and denote the law of τY 0(τB0(x)) by ν¯x. Thensupx∈Σinfν∈P(Σ)T˜ (δx, ν) ≤ supx∈ΣT (ν¯x, δx) ≤ supx∈ΣE[A(Y 0, τB0(x))] ≤ supΣˆA < +∞.For the hypothesis, T˜ (µ, µ) < +∞, and the condition 5.48, they followsimilarly as in the proof of Theorem 5.6.10. We finally need, in order tosatisfy the hypotheses of Theorem 5.6.3, the existence of a function g ∈ C(Σ)such that lim infn→∞(T˜−n g + nc(T )) belongs to USCb(Σ), where we recallthat T˜− is given byT˜−g(x) = −T+(−g)(x) = sup{E[g(τY 0(τB0(x)))−A(Y 0, τB0(x))] ; Y 0 ∈ Σ∗τB0 (x)}188Again, similar to the deterministic case, we can take g ≡ 0. ThenT˜−n (0)(x) = − inf{n−1∑k=0E[A(Y k, τBk(Y¯ k)] ; Y k ∈ Σ∗τBk(Y¯ k)}where Y¯ k is defined recursively via Y¯ 0 = x, Y¯ k = τY k−1(τBk−1(Y¯k−1)),k ≥ 1. At the same time,c(T ) = infµT (µ, µ) = limn→∞1ninfµ,νTn(µ, ν)= limn→∞ infµ,ν1nn−1∑i=0inf{E[A(Y i, X¯i)]}= limn→∞ infµ,ν infX0∼µ,X1∼ν∫ΣˆAdµ(Yi)in= infµˆ∈M0∩P(Σˆ)∫ΣˆAdµˆ.HenceT˜−n (0)(x)+nc(T ) = n infµˆ∈M0∩P(Σˆ)∫ΣˆAdµˆ−inf{n−1∑k=0E[A(Y k, τBk(Y¯ k)] ; Y k ∈ Σ∗τBk(Y¯ k)}.The hypothesis of the theorem ensures that at the liminf as n→∞, we get acontinuous function. Therefore we have satisfied the hypotheses of Theorem5.6.3, and conclude the existence of a h ∈ USC(Σ) such thatT˜−h(x) + c(T˜ ) = h(x), ∀x ∈ Σ,which since T˜−h = −T+(−h) and c(T˜ ) = c(T ), impliesT+(−h)− c(T ) = −h,or with g := −h,T+g(x)− c(T ) = g(x). (5.67)189Replacing x with σ(x) for x ∈ Σ in equation (5.67), we have(T+g)(σ(x))− g(σ(x)) = c(T )Recalling the definition of T+ and the martingale assumption (5.64), we canwritec(T ) = (T+g)(σ(x))− g(σ(x)) = infY 0{E[g(τY 0(τB0(σ(x))) +A(Y 0, τB0(σ(x)))]} − g(σ(x))= infY 0{E[g(τY 0(x)) +A(Y 0, x)]} − g(σ(x))= infy∈Σ∗x{g(τy(x))− g(σ(x)) +A(y, x)}= infy∈Σ∗x{Dyg(x) +A(y, x)} (5.68)Finally, the corresponding Man˜e´ constant is given byc(T ) = infµT (µ, µ) = limn→∞1ninfµ,νTn(µ, ν)= limn→∞ infµ,ν1nn−1∑i=0inf{E[A(Y i, X¯i)]}= limn→∞ infµ,ν infX0∼µ,X1∼ν∫ΣˆAdµ(Yi)in= infµˆ∈M0∩P(Σˆ)∫ΣˆAdµˆ.Since M0 ⊂ N0, it follows that c(T ) ≥ inf µˆ∈N0∩P(Σ)∫ΣˆAdµˆ. In view ofthe duality (5.65) together with (5.68), this implies that this inequality isactually an equality, hence (5.66) holds and concludes the proof.Remark 5.6.16. Similar to the deterministic case, our conjecture is thatwhen A is Ho¨lder continuous, the hypothesis of Theorem 5.6.15 holds.190Bibliography[1] J. Alibert, G. Bouchitte, and T. Champion. A new class of costs foroptimal transport planning. European Journal of AppliedMathematics, 30(6):1229–1263, 2019. → pages 13, 33, 34, 41, 107[2] H. Attouch, G. Buttazzo, and G. Michaille. Variational analysis inSobolev and BV spaces: applications to PDEs and optimization,volume 17. Society for Industrial and Applied Mathematics,Philadelphia, second edition, 2014. ISBN 1611973473;9781611973471;.→ page 8[3] J. Backhoff-Veraguas, M. Beiglbo¨ck, and G. Pammer. Existence,duality, and cyclical monotonicity for weak transport costs. Calculusof Variations and Partial Differential Equations, 58(6):1–28, 2019. →pages 33, 34, 70[4] A. Barton and N. Ghoussoub. Dynamic and stochastic propagation ofthe Brenier optimal mass transport. European Journal of AppliedMathematics, 30(6):1264–1299, 2019. → pages 67, 68[5] P. Bernard and B. Buffoni. Optimal mass transportation and Mathertheory. Journal of the European Mathematical Society, 9(1):85–121,2007. → pages 30, 31, 85, 102, 134, 144[6] P. Bernard and B. Buffoni. Weak KAM pairs and Monge-Kantorovichduality. Advanced Studies in Pure Mathematics, 47(2):397–420, 2007.→ pages 30, 85, 102, 103, 104, 105, 122, 132, 134, 144[7] D. P. Bertsekas and S. E. Shreve. Stochastic optimal control: thediscrete time case, volume 139. Academic Press, New York, 1978. →page 25191[8] S. G. Bobkov and F. Go¨tze. Exponential Integrability andTransportation Cost Related to Logarithmic Sobolev Inequalities.Journal of Functional Analysis, 163(1):1–28, 1999. → pages 73, 74[9] R. Bowen and J. Chazottes. Equilibrium states and the ergodic theoryof Anosov diffeomorphisms, volume 470. Springer, Berlin, 2nd rev.edition, 2008. → pages 133, 170[10] M. Bowles and N. Ghoussoub. Mather measures and ergodicproperties of kantorovich operators associated to general masstransfers, 2019. → pages v, 13[11] Y. Brenier. The least action principle and the related concept ofgeneralized flows for incompressible perfect fluids. Journal of theAmerican Mathematical Society, 2(2):225–255, 1989.[12] Y. Brenier. Polar factorization and monotone rearrangement ofvector-valued functions. Communications on Pure and AppliedMathematics, 44(4):375–417, 1991. → page 32[13] Y. Brenier. The dual least action problem for an ideal, incompressiblefluid. Archive for Rational Mechanics and Analysis, 122(4):323–351,1993.[14] Y. Brenier. Minimal geodesics on groups of volume-preserving mapsand generalized solutions of the Euler equations. Communications onPure and Applied Mathematics, 52(4):411–452, 1999.[15] F. Cagnetti, D. Gomes, and H. Tran. Aubry-Mather measures in thenon convex setting. 2010. → page 85[16] G. Choquet, J. E. Marsden, T. Lance, and S. S. Gelbart. Lectures onanalysis. W.A. Benjamin, New York, 1969. ISBN 9780805369601. →page 27[17] K. J. Ciosmak. Optimal transport and Choquet theory. 2020. → page25[18] D. Cordero-Erausquin and B. Klartag. Moment measures. Journal ofFunctional Analysis, 268(12):3834–3866, 2015. → pages 80, 81[19] L. Evans. A survey of partial differential equations methods in weakKAM theory. Communications on Pure and Applied Mathematics, 57(4):445–480, 2004. → page 85192[20] A. Fathi. Weak KAM Theorem in Lagrangian Dynamics. Version X,2018. → pages 32, 89, 144, 145[21] A. Fathi. Weak KAM Theory : The connection betweenAubry-Mather theory and viscosity solutions of the Hamilton-Jacobiequation. 2018. → page 85[22] M. Fathi, N. Gozlan, and M. Prodhomme. A proof of the Caffarellicontraction theorem via entropic regularization, 2019. → page 65[23] A. Figalli and L. Rifford. Aubry sets, Hamilton-Jacobi equations, andthe Man˜e´ Conjecture, 2012. → page 85[24] J. Fontbona, N. Gozlan, and J. Jabir. A variational approach to sometransport inequalities. Annales de L’Institut Henri Poincare´ -Probabilites et Statistiques, 53(4):1719–1746, 2017. → page 71[25] E. Garibaldi. Ergodic Optimization in the Expanding Case: Concepts,Tools and Applications. Springer International Publishing, Cham, 1st2017 edition, 2017. ISBN 3319666436. → pages 23, 172[26] E. Garibaldi and A. Lopes. On the Aubry–Mather theory for symbolicdynamics. Ergodic Theory and Dynamical Systems, 28(3):791–815,2008. → pages 175, 176, 182, 183[27] I. Gentil, C. Le´onard, and L. Ripani. About the analogy betweenoptimal transport and minimal entropy. Annales de la Faculte´ desSciences de Toulouse. Mathe´matiques, 3:569–600, 2017. → page 47[28] N. Ghoussoub, Y.-H. Kim, and A. Z. Palmer. Optimal transport withcontrolled dynamics and free end times. SIAM Journal on Controland Optimization, 56(5):3239–3259, 2018. → page 51[29] N. Ghoussoub, Y.-H. Kim, and T. Lim. Structure of optimalmartingale transport plans in general dimensions. Annals ofProbability, 47(1):109–164, 2019. → page 46[30] N. Ghoussoub, Y.-H. Kim, and T. Lim. Optimal brownian stoppingwhen the source and target are radially symmetric distributions. 2019.→ pages 27, 50[31] N. Ghoussoub, Y.-H. Kim, and A. Z. Palmer. A solution to theMonge transport problem for Brownian martingales. 2019. → pages49, 50, 51, 126, 127193[32] D. Gomes. Viscosity solution methods and the discrete Aubry-Matherproblem. Discrete and Continuous Dynamical Systems, 13(1):103–116,2005. → page 85[33] N. Gozlan and N. Juillet. On a mixture of Brenier and StrassenTheorems. Proceedings of the London Mathematical Society, 120(3):434–463, 2020. → page 65[34] N. Gozlan and C. Le´onard. Transport inequalities. A survey. MarkovProcesses and Related Fields, 16:635–736, 2010. → pages 72, 73[35] N. Gozlan, C. Roberto, and P.-M. Samson. From concentration tologarithmic Sobolev and Poincare´ inequalities. Journal of FunctionalAnalysis, 260(5):1491–1522, 2011. → pages 20, 57[36] N. Gozlan, C. Roberto, P.-M. Samson, and P. Tetali. Kantorovichduality for general transport costs and applications. Journal ofFunctional Analysis, 273(11):3327–3405, 2017. → pages 13, 33, 45, 65[37] P. Henry-Labordere. Model-free hedging: A martingale optimaltransport viewpoint. Chapman & Hall, London, 1 edition, 2017. ISBN1351666231. → page 46[38] M. Ledoux. The concentration of measure phenomenon, volume 89;no.89.;. American Mathematical Society, Providence, R.I, 2001. ISBN0821828649;9780821828649;. → page 73[39] P. Lions, G. Papanicolaou, and S. Varadhan. Homogenization ofHamilton-Jacobi equations. 1987. → page 88[40] D. Martino. personal communication. → page 172[41] K. Marton. Bounding d¯-distance by informational divergence: Amethod to prove measure concentration. The Annals of Probability, 24(2):857–866, 1996. ISSN 00911798. URLhttp://www.jstor.org/stable/2244952. → pages 44, 72[42] K. Marton. A measure concentration inequality for contractingMarkov chains. Geometric and Functional Analysis, 6(3):556–571,1996. → pages 44, 72[43] J. N. Mather. Action minimizing invariant measures for positivedefinite Lagrangian systems. Mathematische Zeitschrift, 207(1):169–207, 1991. → page 144194[44] B. Maurey. Some deviation inequalities. Geometric and FunctionalAnalysis, 1(2):188–197, 1991. → page 84[45] T. Mikami and M. Thieullen. Duality theorem for the stochasticoptimal control problem. Stochastic Processes and their Applications,116(12):1815–1835, 2006. → page 49[46] R. R. Phelps. Convex functions, monotone operators anddifferentiability, volume 1364;. Springer-Verlag, New York;Berlin;, 2ndedition, 1989;1993;. ISBN 3540507353. → page 70[47] V. Roos. Variational and viscosity operators for the evolutiveHamilton-Jacobi equation. Communications in ContemporaryMathematics, 2017. → page 13[48] F. Santambrogio. Optimal Transport for Applied Mathematicians:Calculus of Variations, PDEs, and Modeling, volume 87. SpringerInternational Publishing, Cham, 1st 2015.;1st 2015; edition, 2015. →pages 28, 148[49] F. Santambrogio. Crowd motion and evolution PDEs under densityconstraints. ESAIM: ProcS, 64:137–157, 2018.doi:10.1051/proc/201864137. URLhttps://doi.org/10.1051/proc/201864137. → page 58[50] E. Schechter. Handbook of analysis and its foundations. AcademicPress, San Diego, 1997;1996;. ISBN 9780126227604. → page 129[51] Y. G. Sinai. Dynamical Systems II: Ergodic Theory with Applicationsto Dynamical Systems and Statistical Mechanics, volume 2. SpringerBerlin Heidelberg, Berlin, Heidelberg, 1989. ISBN 9783662067888. →pages 157, 179[52] M. Sion. On general minimax theorems. Pacific journal ofmathematics, 8(1):171–176, 1958. → page 39[53] A. V. Skorokhod. Studies in the theory of random processes, volume7021;. Addison-Wesley Pub. Co, Reading, Mass, 1965. → page 50[54] A. Sorrentino. Lecture notes on Mather’s theory for Lagrangiansystems. 2010. → page 85[55] A. Sorrentino. Action-minimizing Methods in Hamiltonian Dynamics:An Introduction to Aubry-Mather Theory. Princeton University Press,1952015. ISBN 9780691164502. URLhttp://www.jstor.org/stable/j.ctt1b9x11w. → page 85[56] V. Strassen. The Existence of Probability Measures with GivenMarginals. The Annals of Mathematical Statistics, 36(2):423–439,1965. → page 25[57] M. Talagrand. Transportation cost for Gaussian and other productmeasures. Geometric and Functional Analysis, 6(3):587–600, 1996. →page 72[58] Treschev, Dmitry and Zubelevich, Oleg. Introduction to the KAMTheory. In Introduction to the Perturbation Theory of HamiltonianSystems, pages 23–58, Berlin, Heidelberg, 2010. Springer BerlinHeidelberg. ISBN 978-3-642-03028-4.doi:10.1007/978-3-642-03028-4 2. URLhttps://doi.org/10.1007/978-3-642-03028-4 2. → page 88[59] C. Villani. Topics in optimal transportation, volume 58. AmericanMathematical Society, Providence, R.I, 2003. ISBN 9780821833124.→ pages 28, 29, 163, 172[60] C. Villani. Optimal transport: old and new, volume 338. Springer,Berlin, 2009;2008;. ISBN 9783540710493. → pages 8, 28[61] C. E. Wayne. An introduction to KAM theory. In Dynamical Systemsand Probabilistic Methods in Partial Differential Equations, pages3–29, 1994. → page 88196
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Mathematics
Vancouver : University of British Columbia Library
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Linear transfers, Kantorovich operators, and their ergodic properties
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