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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 .

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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

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