Let (W, µ, H ) be an abstract Wiener space assume two ν i , i = 1, 2 probabilities on (W, B(W )) 1 . We give some conditions for the Wasserstein distance between ν 1 and ν 2 with respect to the Cameron-Martin spaceto be finite, where the infimum is taken on the set of probability measures β on W × W whose first and second marginals are ν 1 and ν 2 . In this case we prove the existence of a unique (cyclically monotone) map T = I W + ξ , with ξ : W → H , such that T maps ν 1 to ν 2 . Moreover, if ν 2 µ 2 , then T is stochastically invertible, i.e., there exists S : W → W such that S • T = I W ν 1 a.s. and T • S = I W ν 2 a.s. If, in addition, ν 1 = µ, then there exists a 1-convex function φ in the Gaussian Sobolev space D 2,1 , such that ξ = ∇φ. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from µ can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W .

Abstract1 Inspired by the fundamental work of T.J. Lyons (loc.cit), we develop a theory of curvilinear integrals along a new kind of enriched paths in IR d . We apply these methods to the fractional Brownian Motion, and prove a support theorem for SDE driven by the Skorohod fBM of Hurst parameter H > 1/4.

We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting. Academic Press

Abstl~et. We construct a 'Hausdorff measure' of finite co-dimension on the Wiener space. We then extend the Federer co-area Formula to this Wiener space for functions with the sole condition that they belong to the first Sobolev space. An explicit formula for the density of the images of the Wiener measure under such functions follows naturally from this. As a corollary, this yields a new and easy proof of the Kr6e-Watanabe theorem concerning the regularity of the images of the Wiener measure.Mathematies Subject Cl~aifieations (1991). 28A50, 28C20, 46E35, 46F05, 46F25, 46G12, 60(315, 60H07.

A non-commutative version of the sewing lemma [1] is proved, with some applications.1 Definition : We say that a function V (t) defined on [0, T [ is a control function if it is non decreasing, V (0) = 0 and n≥1 V (1/n) < ∞.As easily seen, this is equivalent to the propertyfor every t ≥ 0. For example, t α and t/(log t −1 ) α with α > 1 are control functions.Observe that we havea control function. Then there exists a unique function ϕ(t) on [0, T [, up to an additive constant, such that |ϕ(b) − ϕ(a) − µ(a, b)| ≤ V (b − a) Proof : Put µ ′ (a, b) = µ(a, c) + µ(c, b) for c = (a + b)/2, and µ (n+1) = µ (n)′ . We easily get for n ≥ 0 a) converges, and the sequence µ (n) (a, b) converges to a limit u(a, b). For c = (a + b)/2 we haveWe say that u is midpoint-additive. Now, we prove that u is the unique midpoint-additive function with the inequalityand by induction |v(a, b) − u(a, b)| ≤ 2 n K.V [2 −n (b − a)] which vanishes as n → ∞ as mentioned above. Let k be an integer k ≥ 3, and take the function