Table Of Content

Optimal transport on Wiener space with different norms Vincent NOLOT 2 1 Institut de Mathématiques de Bourgogne, 0 Université de Bourgogne, 21078 Dijon, France. 2 [email protected] p e S September 7, 2012 6 ] R Abstract P InthispaperwestudytwobasicfactsofoptimaltransportationonWienerspaceW. . h OurfirstaimistoanswertotheMongeProblemontheWienerspaceendowedwiththe t Sobolevtypenorm . p withp 1(casesp=1andp>1areconsideredapart). The a kkk,γ ≥ m second one is to prove 1 convexity (resp. 1/C2 convexity) along (constant speed) − k,γ− [ geodesics of relative entropy in ( 2(W),W2), where W is endowed with the infinite P norm (resp. with . k,γ), and W2 is the 2 distance of Wasserstein. 5 kk − v 8 1 Introduction 3 9 1 We are interested in two problems in optimal transportation on Wiener space. We . referto a recentworkofAmbrosioandGigli[3]for asurveyandbasic toolsofoptimal 1 transport theory. 1 1 1 AtfirstweanswertotheMongeProblemontheWienerspacerelativelytothecost v: (introduced by Airault and Malliavin in [1]) k.kpk,γ defined as : i X 1 1 (w(t) w(s))2k 1/2k r w := − dtds , a k kk,γ t s1+2kγ (cid:18)Z0 Z0 | − | (cid:19) for suitable parameters k,γ. Monge Problem is largely considered in several settings since for years. Nowadays therearealotofmeanstoprovetheexistenceofanoptimalmapresolvingthisProblem, which are summarized in [2]. Recently, there was considerableadvances concerning Monge Problem in Rn. First in 1996, Gangbo and McCann solved Monge Problem in [15] when the cost is strictly convex. Then people are interesting in the case of different norms on Rn. Indeed the problem becomes more difficult since a norm is never strictly convex. In 2003, Monge 1 Introduction Problem was solved when the cost was a crystalline norm by Ambrosio, Kirchheim and Pratelli in [5]. When the cost is a general norm, Monge Problem was solved independently by Champion, De Pascale in [9] and by Caravennain [7] in 2010. All of these latter cases we lose unicity of optimal map. In this paper, we turn our attention on the Wiener space, an infinite dimensional space. The Monge Problem in the Wiener space was solved by Feyel and Ustunel in [14] for the cost .2 induced by Cameron-Martin norm . which is Hilbertian. ||H ||H With this cost, we haveunicity of optimalmap. Strategyof latter authorswasto pass by finite dimensional approximations of Wiener space : the aim being to reduce the Monge Problem on Wiener space onto finite dimensional spaces and applying known results. Then they used a selection theorem to go back up on Wiener space. Then MongeProblemwassolvedby Cavallettiin[8]forthe cost . , againpassingbyfinite H || dimensional approximations. We are interested to endow the Wiener space (W,H,µ) (where µ is the Wiener measure) with two other natural norms : the infinite norm . and the Sobolev type ∞ || norm . . For the first one, Monge Problem is still open, and we can expect not to k,γ kk have unicity of optimal map, providing it exists somehow. Here the norm considered . is notHilbertianandis weakerthanthe Cameron-Martinnorminsensethatfor k,γ kk some C >0 : k,γ x C x for all x W. (1.1) k,γ k,γ H k k ≤ | | ∈ Let us emphasize that the right hand side is equal to infinite µ almost everywhere, − because of zero measure of Cameron-Martin space. Nevertheless our norm . sat- k,γ kk isfies suitableconditionspresentedinsection2.1. The firstaimofourpaperisto solve Monge Problem for the cost . p : kkk,γ inf G(x) x p dρ (x). (1.2) G#ρ0=ρ1ZW×W k − kk,γ 0 In other words we will establish following theorem : Theorem 1.1 Let ρ and ρ be two measures on W satisfying condition (1.3) below. 0 1 1. If p>1 and ρ is absolutely continuous with respect to µ, then there exists a map 0 T unique up to a set of zero measure for ρ , which minimizes (1.2). Moreover 0 there is a unique optimal transference plan between ρ and ρ relatively to the 0 1 cost . p , which is exactly (Id T) ρ . kkk,γ × # 0 2. If p = 1 and both ρ and ρ are absolutely continuous with respect to µ, then 0 1 there exists an optimal transference plan Π between ρ and ρ relatively to the 0 1 cost . , such that Π is concentrated on a graph of some map T minimizing k,γ kk (1.2). The secondpurpose of this paper is to find a lowerbound of Ricci curvaturein the sense of Sturm in [19], for Wiener space endowed with the infinite norm . or the ∞ || norm . . Itisreliedto weakK convexityalonggeodesicofrelativeentropy. More k,γ kk − precisely we will prove : 2 Introduction Theorem 1.2 Ifρ andρ are probability measureson W both absolutely continuous 0 1 with respect to µ, then there exists some (constant speed) geodesic ρ induced by an t optimal transference between ρ and ρ such that : 0 1 Kt(1 t) Ent (µ ) (1 t)Ent (ρ )+tEnt (ρ ) − W2 (ρ ,ρ ) t [0,1]. µ t ≤ − µ 0 µ 1 − 2 2,∞ 0 1 ∀ ∈ For the infinite norm, we will see that K equals to 1, and for the Sobolev type norm . that K equals to 1/C2 . Precise that Lott-Villani introduced in [17] a kkk,γ k,γ stronger notion of weak K convexityof relative entropy. Indeed in its definition, it is − required that the property above holds for all (constant speed) geodesics. So in many cases weak K convexity and K convexity coincide, provided that there is unicity of − − geodesic between two given measures (this is the case for non branching spaces). An example it fails when optimal coupling is not unique. Now let us briefly summarize the following sections. Throughout section 2, we resolve the Monge Problem in the Wiener space with the cost . p , in different ways according to parameter p. Among of these, there is kkk,γ a direct method : in general settings, the support of an optimal transference plan Π (between two probability measures and relatively to a cost function c) is included in c subdifferential of a c convex function (called potential of Kantorovich) φ. It leads − − to the following system : φc(y) φ(x) = c(x,y) Π almost everywhere − − φc(y) φ(x) c(x,y) everywhere (cid:26) − ≤ Andthissystemcanbesolveddirectlywhenthecostcandthepotentialφaredifferen- tiable, so long as c(x,.) is injective, as it is explained in Villani’s book [20]. This is x ∇ thecasewhenp>1. Butthismethodfailswhenp=1. Inthelattercaseweheadfora recent paper of Bianchini and Cavalletti [6] where the authors resolve Monge Problem innonbranchinggeodesicmetricspaces. ItturnsoutthatWienerspaceendowedwith the norm . is a such space. Simply we will verify suitable conditions. k,γ kk Section 3 is devoted to establish the K convexity along geodesic of relative en- − tropy (w.r.t. Wiener measure) on the Wiener space endowed with the infinite norm (K = 1), then with the Sobolev type norm . (K = 1/C2 ). This time we will kkk,γ k,γ processbyfinitedimensionalapproximationasFang,ShaoandSturmin[12],whohave treated the case of the norm . . This part requires Wasserstein distance, which is H || definedbelow. Ourmaincontributionconsistsinestablishingresultswithoutapplying powerfull tools like Gromov-Hausdorff convergence(see [17]) or D convergenceintro- − duced by Sturm in [19]. In the language of latter authors, we can say that (W, . ) ∞ || is a CD(1, ) space (satisfies such curvature-dimension condition) and (W, . ) k,γ ∞ kk is a CD(1/C2 , ) space. Such conditions imply a lot of important results over k,γ ∞ Wienerspaces. Asconsequencesoverspaces(W, . ), (W, . ),wecanquoteBrunn- ∞ k,γ || kk Minkowski, Bishop-Gromovor also Log-Sobolev inequalities (see [3]). 3 1.1 Settings Introduction 1.1 Settings OurambiantspacewillbetheclassicalWienerspace(W,H,µ)=( ([0,1],R),H(R),µ) 0 where H(R) := h : [0,1] R; h(t) = th˙(s)ds and h˙ L2(C[0,1]) and µ is the { −→ 0 ∈ } Wiener measure. H(R) is a Hilbert space with the inner product : R (h,g) := h˙(t)g˙(t)dt. H Z[0,1] Important facts are that H is dense in W with respect to the uniform norm, and moreover µ(H)=0. Given two Borel measures ρ and ρ on W, let us state a condition of preserving 0 1 mass, necessary for all our discussion : dρ = dρ <+ . (1.3) 0 1 ∞ ZW ZW In particular it is satisfied when ρ and ρ are both probability measures. When (1.3) 0 1 is satisfied, we will consider many times the following Monge-KantorovichProblem inf (Π)= inf x y p dΠ(x,y), (1.4) Π∈Γ(ρ0,ρ1)Ip Π∈Γ(ρ0,ρ1)ZW×W k − kk,γ which is a relaxed problem of Monge Problem (1.2) stated above. At the moment of the cost c is continuous and the ambiant space is Polish, there is always an existing minimizerfortheMonge-KantorovichProblem(1.4)(seeforexample[4]). Asuchmini- mizerwillbecalledanoptimalcoupling oroptimaltransferenceplan betweenρ andρ . 0 1 We endow the space W with the following norm : 1 1 (w(t) w(s))2k 1/2k w := − dtds for w W, k kk,γ t s1+2kγ ∈ (cid:18)Z0 Z0 | − | (cid:19) such that 0 < γ < 1/2, and 2 < 1+2kγ < k where k is an integer. In fact this is a pseudo-norm over W since it can take infinite value. For this reason, we consider Wˆ := w W; w < . It is well known that µ(Wˆ) = 1 hence for a sake k,γ { ∈ k k ∞} of notation, we will write W = Wˆ. (W, . is a separable Banach space, and all k,γ kk measures considered in the sequel will be Borel with respect to the topology induced by the norm . . H is still dense in (W, . ). We can write : k,γ k,γ kk kk 1 1 1 2 x(t) x(s)2(k−1) w 2k x˙(ξ)1 dξ | − | dtds k kk,γ ≤ | | s<t t s1+2kγ Z0 Z0 (cid:18)Z0 (cid:19) | − | 1 1 1 x(t) x(s)2(k−1) x˙(ξ)2dξ t s | − | dtds ≤ | | | − | t s1+2kγ (cid:18)Z0 (cid:19)Z0 Z0 | − | C2k x2k, ≤ k,γ| |H 1/2k where C := 1 1 t sk−1−2kγdtds . k,γ 0 0 | − | (cid:16) (cid:17) R R 4 1.1 Settings Introduction Beforetocontinue,letusexplainhowwecandecomposetheclassicalWienerspace in finite dimensional spaces. Consider the projections π :W W defined as : n −→ k k k+1 k k k+1 π (x)(t):=x +2n t x x , if t , . n 2n − 2n 2n − 2n ∈ 2n 2n (cid:18) (cid:19) (cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21) At a continuous path, π associatesits affine part. Denote by V :=π (W). We know n n n thatV isspannedbytheHaarfunctionsandthereforehasdimension2n andthenwill n be identified with RN. Denote by V⊥ the subset of W which is image of W by the map I π . Since n W − n π is a projection, one can write W =V V⊥. n n⊕ n An important fact is the following : the image measure of the Wiener measure µ by π is the standard Gaussian measure on V . In other words (π ) µ = γ , where n n n # d the indice d is the dimension of V i.e 2n =d. For a sake of simplicity, we will denote n (π ) µ=γ instead of γ . n # n d RecallthattheSobolevspace overtheWienerspaceDp(W)isthesetofallµ measurable 1 − functionF Lp(W,µ)suchthatthereexists F Lp(W,H)wherewehaveµ almost ∈ ∇ ∈ − surely : F(w+ǫh) F(w) ( F(w),h) =D F(w):= lim − in Lp(W,µ) h H. H h ∇ ǫ→0 ǫ ∀ ∈ We will denote D∞(W):= Dp(W) and E(F):=E (F). 1 1 µ p>0 \ Insection3wewillneed2 Wassersteindistancebetweentwoprobabilitymeasures − ρ and ρ on a measurable space W, defined as : 0 1 W2 (ρ ,ρ ):= inf c(x,y)2dΠ(x,y) (1.5) 2,c 0 1 Π∈Γ(ρ0,ρ1)Z where Γ(ρ ,ρ ) is the set of coupling measures between ρ and ρ i.e the measures on 0 1 0 1 W W with marginals ρ and ρ . Moreover we will denote by Γc(ρ ,ρ ) the set of × 0 1 0 0 1 optimalcouplings forc2 orequivalently the set ofcouplingswhichrealizeminimum on 2 Wassersteindistance inducedbythe cost c. Asufficientcondition(but notnatural) − for that the Wasserstein distance is that ρ and ρ have second finite moments. To 0 1 justify terminology, notice that W is well a distance so long as c is a distance on W 2,c and in this case the space (W):= ρ probability measure on W; c(x,x )2dρ(x)< for some x W 2 0 0 P ∞ ∈ (cid:26) ZW (cid:27) endowed with W is a metric space. 2,c In all this paper, Pr (with 1 i N) stand for the projections onto the i th i ≤ ≤ − component : Pr :X X ...X X , i 1 2 N i × × −→ where N N depends on the context. ∈ 5 Monge Problem on Wiener space with . p kkk,γ 2 Monge Problem on Wiener space with . p k kk,γ We can now focus on the Monge Problem. Notice that (W, . ) is a Polish space k,γ and in addition the cost c(x,y)= x y p being continuousk(kfor all p 1), there is k − kk,γ ≥ always an existing measure Π Γ(ρ ,ρ ) which attains the minimum in the Monge- 0 1 ∈ KantorovichProblem (1.4). For ρ , ρ two measures on W satisfying (1.3), recall that the Monge problem 0 1 between ρ and ρ consists of finding an optimal map T : W W which pushes ρ 0 1 0 −→ forwards to ρ and minimizes the quantity : 1 c(x,G(x))dρ (x), 0 ZW among all push-forward maps G (i.e. G ρ (E) := ρ (G−1(E)) = ρ (E) for all Borel # 0 0 1 E subset of W). The usual strategy is to use characterization of optimal coupling, with the help of KantorovichPotentials. Hence we need the concept of c convexity. − Definition 2.1 Let ϕ:W R. We say that ϕ is c convex if : −→ − ϕ(x)= sup (ϕc(y) c(x,y)) x W. − ∀ ∈ y∈W where ϕc, called c transform of ϕ, is defined as : − ϕc(y)= inf (ϕ(x)+c(x,y)) y W. x∈W ∀ ∈ Notice that our cost c does not take an infinite value, so c convex functions are − real-valued well. Itiswellknownthatanyoptimalcouplingis c cyclicallymonotonei.e. its support − (defined as the smaller closed subset of W W having full Π measure) is c cylically × − − monotone, namely : for all N N and (x ,y ),...,(x ,y ) Supp(Π) we have : 1 1 N N ∈ ∈ N N c(x ,y ) c(x ,y ), i i i i+1 ≤ i=1 i=1 X X with y :=y . N+1 1 Rather there is equivalence (in our case) between optimality and c cyclical mono- − tonicity (see e.g. [3]). In particular if Supp(Π) is c cyclically monotone then any coupling Π˜ such that Supp(Π˜) Supp(Π) is also c cy−clically monotone. ⊂ − Let us precise what are the difficulties we have met. When p =1, c(x,y)= x y is the considered norm on W. Hence if a map k,γ • k − k ϕ is c convex then it is also 1 Lipschitz, hence H Lipschitz. Indeed : − − − ϕ(x+h) ϕ(x) h C h h H x W. k,γ k,γ H | − |≤k k ≤ | | ∀ ∈ ∀ ∈ Inthiscase,wehaveaversionofRademachertheorem(provedin[10]andrecalled in appendice for a sake of completeness) on Wiener space, for such H Lipschitz − 6 2.1 Properties of . Monge Problem on Wiener space with . p kkk,γ kkk,γ functions. With the remark above,it leads to c convexfunctions onthe Wiener − space are almost surely differentiable. But the difficulty in this case is that the cost, being a norm, is not strictly convex, so we do not have injectivity of y c(x,y), therefore our method is not available. Nevertheless we shall x 7−→ ∇ use an other method to solve Monge Problem. Indeed thanks to the part 2. of Lemma(2.1),(W, . )isageodesicnonbranchingspace(i.e. geodesicscannot k,γ kk bifurcate). So we can apply method, detailed in [6]. When p > 1, c(x,y) = x y p becomes strictly convex, we get this time • k − kk,γ the injectivity of c(x,.). But we lose the H Lipschitz property of c convex x ∇ − − functions. Indeed if ϕ is such function we can write : ϕ(x) ϕ(y) x ξ p y ξ p | − | ≤ |k − kk,γ −k − kk,γ| x y M , k,γ ξ ≤ k − k where the latter constant M depends on ξ and cannot be bounded. However ξ we will see that in this case c convex functions (hence potentials) are locally − H Lipschitz. Since differentiability is a local property, we should be able to − apply Rademacher theorem again. Beforewecontinue,letussetpropertiesoftheconsiderednorm,whichwillbeuseful for the sequel. 2.1 Properties of . k,γ k k We give two ingredients that will be essential for the sequel. Lemma 2.1 If we denote by F˜ : W R the map F˜(w) = w , then we have + k,γ −→ k k the following properties : 1. F˜ admits a gradient F˜(w) belonging to W⋆ for all w W 0 , where W⋆ is the dual of W. Moreo∇ver F˜p is everywhere differentiable∈for a\l{l p}>1. 2. F˜ is a norm such that its unit ball is strictly convex. The first part of the proof is inspired from [11]. Proof : 1. Firstwe show the propertyfor F :=F˜2k. Takeh W, we canwrite for w W and ∈ ∈ ǫ>0 : 1 1 ((w(t) w(s))+ǫ(h(t) h(s)))2k F(w+ǫh)= − − dtds. t s1+2kγ Z0 Z0 | − | And taking the derivative at ǫ = 0, it is clear that lim F(w+ǫh)−F(w) exists and ǫ→0 ǫ moreover : 1 1 w(t) w(s)2k−1 D F(w) 2k | − | h(t) h(s)dtds | h | ≤ t s1+2kγ | − | Z0 Z0 | − | w(t) w(s)2k−1 h(t) h(s) 2k | − | | − | dtds ≤ t s(1+2kγ)(2k−1)/(2k) t s(1+2kγ)/(2k) Z[0,1]2 | − | | − | 7 2.2 The case p > 1. Monge Problem on Wiener space with . p kkk,γ and now applying Hölder’s inequality, we get : (2k−1)/(2k) 1/(2k) w(t) w(s)2k h(t) h(s)2k D F(w) 2k | − | dtds | − | dtds | h | ≤ Z[0,1]2 |t−s|1+2kγ ! Z[0,1]2 |t−s|1+2kγ ! = 2k w 2k−1. h . k kk,γ k kk,γ Hence h D F(w) is a bounded operatoron W for allw W. It leads to existence h 7−→ ∈ of a gradient F(w) which belong to the dual space W⋆ H⋆ =H (by (1.1)). Since F˜ = F1/(2k),∇its gradient satisfies F˜(w) = F1/(2k)−1(w)⊂F(w) and in particular w ∇ ∇ must not be equal to zero. Since F˜ is differentiable out of 0 it is just a remark to see that for any p>1, F˜p { } is differentiable everywhere over (W, . ). k,γ kk 2.ThisproofisthesameastheproofofMinkowski’sinequality. Indeedifw , w W 1 2 ∈ and η (0,1) then we have : ∈ (1 η)(w (t) w (s))+η(w (t) w (s))2k (1 η)w +ηw 2k = | − 1 − 1 2 − 2 | dtds k − 1 2kk,γ t s1+2kγ Z[0,1]2 | − | = (1 η)(w (t) w (s))+η(w (t) w (s)) 1 1 2 2 | − − − | Z[0,1]2 (1 η)(w (t) w (s))+η(w (t) w (s))2k−1 1 1 2 2 | − − − | dtds × t s1+2kγ | − | (1 η)w (t) w (s) (1 η)(w (t) w (s))+η(w (t) w (s))2k−1 1 1 1 1 2 2 − | − || − − − | dtds ≤ Z[0,1]2 |t−s|(1+2kγ)/(2k) |t−s|(1+2kγ−21k−γ) η w (t) w (s) (1 η)(w (t) w (s))+η(w (t) w (s))2k−1 2 2 1 1 2 2 + | − | | − − − | dtds Z[0,1]2 |t−s|(1+2kγ)/(2k) |t−s|(1+2kγ−21k−γ) ((1 η) w +η w ) (1 η)w +ηw 2k 1−1/2k. ≤ − k 1kk,γ k 2kk,γ k − 1 2kk,γ (cid:0) (cid:1) The two inequalitiesabovearerespectivelytriangleinequality andHolder’sinequality, and are in fact equality if and only if w and w are almost everywhere colinear. This 1 2 leads to the strict convexity of our norm. Conditions on parameters (p,γ) are sufficient to have F D∞(W), as it is shown in [11]. Notice that the (Gateaux) differentiability of F˜ exis∈ts in1 the direction of W, hence in particular in the direction of H. 2.2 The case p > 1. Throughout this subsection, the cost is c(x,y)= x y p , with p>1. k − kk,γ We follow Fathi and Figalli in [13] to get around the fact that c convex functions − are not 1 Lipschitz with respect to . , but nevertheless are locally Lipschitz with k,γ − kk restrictiontosuitablesubsets. Thekeyargumentisthatthesupofafamilyofuniformly . Lipschitz functions, is also . Lipschitz. k,γ k,γ kk − kk − 8 2.2 The case p > 1. Monge Problem on Wiener space with . p kkk,γ Theorem 2.1 Let ρ and ρ be two measures on W satisfying (1.3) and such that 0 1 the first one is absolutely continuous with respect to the Wiener measure µ. Assume (Π) is finite for some Π Γ(ρ ,ρ ). 0 1 I ∈ Then there exists a unique optimal coupling between ρ and ρ relatively to the cost 0 1 c. Moreoever it is concentrated on a graph of some Borel map T :W W unique up −→ to a set of zero measure for µ. Proof : LetΠ Γ(ρ ,ρ )beanoptimalcouplingforc. WeshallshowthatΠisconcentratedon 0 1 ∈ agraphofsomeBorelmap. Itiswellknown(seee.g. [20])thatundercondition (Π)is I finite, since Supp(Π) is c cyclically monotone, there is a c convex map ϕ:W R − − −→ (called Kantorovichpotential) such that ϕc(y) ϕ(x)= x y p Π a.s. − k − kk,γ − Moreover from the definition of c convexity,we also have − ϕc(y) ϕ(x) x y p (x,y) W W. (2.6) − ≤k − kk,γ ∀ ∈ × Since ϕc is finite everywhere, if we consider subsets W := ϕc n for n N then : n { ≤ } ∈ W W and W =W. n n+1 n ⊂ n∈N [ Our cost c(.,y) = . y p is locally . Lispchitz locally uniformly in y. Hence k − kk,γ kkk,γ− for each y W there exists a neighborhood E of y such that ( . z p ) is a ∈ y k − kk,γ z∈Ey uniform family of locally . Lipschitz functions. MoreoverW being separable,we k,γ kk − can find a sequence (yl)l∈N of elements of W such that : E =W. yl l∈N [ Now consider increasing subsets of W : n V :=W ( E ). n n yl l=1 \ [ We can define maps approximating ϕ as follow : ϕ :W W n −→ x sup ϕc(y) x y p . 7−→ −k − kk,γ y∈Vn (cid:16) (cid:17) Notice that ϕ (x)= max sup ϕc(y) x y p . n l=1,...,ny∈Wn∩Eyl(cid:16) −k − kk,γ(cid:17) But since ϕ n on W and . p is bounded from above, ϕ is also bounded ≤ n −kkk,γ n from above. Therefore the sequel (ϕc(y) . y p ) is uniformly locally −k − kk,γ y∈Wn∩Eyl 9 2.2 The case p > 1. Monge Problem on Wiener space with . p kkk,γ . Lipschitzandboundedfromabove. FinallyProposition.4inAppendicesshows k,γ kk − thatϕ beingamaximumofuniformlylocally . Lipschitzfunctions,isalsolocally n k,γ kk − . Lispchitz. We canextend ϕ to a . Lipschitz function everywhereon W k,γ n k,γ kk − kk − still denoted by ϕ . By (1.1), we get : n ϕ (w+h) ϕ (w) C h 2C h w W, h H. n n k,γ H | − |≤ k k ≤ | | ∀ ∈ ∀ ∈ Namely,ϕ isaH Lipschitzfunction. ThankstoRademachertheorem,thereexistsa n − BorelsubsetF ofW withplainµ (henceρ )measuresuchthatforallx F ,ϕ is n 0 n n − − ∈ differentiable at x. Then for each x F := F (which has also plain ρ measure), n n 0 ∈ ∩ − each ϕ is differentiable at x. n By increasing of (V ) , it is clear that ϕ ϕ ϕ everywhere on W. More- n n n n+1 ≤ ≤ over with same argument as in [13], if P := Pr (Supp(Π) (W V )) then ϕ = n 1 ∩ × n |Pn ϕ =ϕ for alll n and alln N. Fix x P F. By definition of P it exists n|Pn l|Pn ≥ ∈ ∈ n∩ n y V such as : x n ∈ ϕc(y ) ϕ (x) = x y p , x − n k − xkk,γ i.e. ϕc(y ) ϕ(x) = x y p . x − k − xkk,γ Subtracting (2.6) with (x′,y ) to the previous equality, we get for all x′ W and x ∈ h H : ∈ ϕ(x) ϕ(x′) x y p x′ y p . − ≥k − xkk,γ −k − xkk,γ Taking x′ =x+ǫh with ǫ>0, h H, dividing by ǫ and taking the limit when ǫ tends ∈ to 0, we get (by linearity in h) : ϕ(x)+ c(x,y )=0. (2.7) x x ∇ ∇ Indeed c(.,y ) is differentiable at x thanks to Proposition 2.1. The strict convexity of x c(x,y)= x y p yields c(x,.) is injective and (2.7) gives : k − kk,γ ∇x y =( c(x,.))−1( ϕ(x))=:T(x), x x ∇ −∇ where ( c(x,.))−1 is the inverse of the map y c(x,y). Notice here that T is x x ∇ 7−→ ∇ uniquely determined. We deduce that Supp(Π) (W V ) is the graph of the map T n ∩ × overP F foralln N. But(P ) and(V ) areincreasingandsuchthat V =W. n∩ ∈ n n n n n n Therefore Supp(Π) is a graph over Pr (Supp(Π)) F with Pr (Supp(Π))= P . 1 ∩ 1 S n n We can extend T onto a measurable map over W as it is explained in [13]. We S obtain Supp(Π) is included in the graphof a measurable map T,unique up to a set of ρ measure. In other words Π=(id T) ρ . 0 # 0 − × We have proved that any optimal coupling is carried by a graph of some map. So if Π , Π Γ(ρ ,ρ ) are optimalfor . then any convexcombinationof Π andΠ 1 2 0 1 k,γ 1 2 ∈ kk is also optimal. Take Π := 1(Π +Π ) be an optimal coupling between ρ and ρ : 2 1 2 0 1 there exists some measurablemap T such thatΠ=(Id T) ρ . Let f be the density # 0 × 10