Probability Theory and Related Fields

, Volume 128, Issue 3, pp 347–385

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

• D. Feyel
• A. S. Üstünel
Article

## Abstract

Let (W,μ,H) be an abstract Wiener space assume two ν i ,i=1,2 probabilities on (W,ℬ(W)). We give some conditions for the Wasserstein distance between ν1 and ν2 with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures β on W×W whose first and second marginals are ν1 and ν2. In this case we prove the existence of a unique (cyclically monotone) map T=I W +ξ, with ξ:WH, such that T maps ν1 to ν2. Moreover, if ν2≪μ, then T is stochastically invertible, i.e., there exists S:WW such that ST=I W ν1 a.s. and TS=I W ν2 a.s. If, in addition, ν1=μ, then there exists a 1-convex function φ in the Gaussian Sobolev space 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.

## Keywords

Monge-Kantorovitch problem Measure transportation Monge-Ampère equation Wiener space Girsanov theorem Transport inequalities

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