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Let ( W,µ, H) be an abstract Wiener space assume two ? i, i=1,2 probabilities on ( W,B( W)). We give some conditions for the Wasserstein distance between ? 1 and ? 2 with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures ß on W× W whose first and second marginals are ? 1 and ? 2. In this case we prove the existence of a unique (cyclically monotone) map T= I W+?, with ?: W? H, such that T maps ? 1 to ? 2. Moreover, if ? 2«µ, then T is stochastically invertible, i.e., there exists S: W? W such that S? T= I W ? 1 a.s. and T? S= I W ? 2 a.s. If, in addition, ? 1=µ, then there exists a 1-convex function f in the Gaussian Sobolev space such that ?=?f. 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.

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space