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

Abstract

Let (W, μ, H) be an abstract Wiener space assume two νi, i = 1, 2 probabilities on (W, B(W))1. We give some conditions for the Wasserstein distance between ν1 and ν2 with respect to the Cameron-Martin space dH(ν1, ν 2) = √infβ ∫W×W |x - y|H2dβ(x, y) 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 = IW + ξ, with ξ: W → H, such that T maps ν1 to ν 2. Moreover, if ν2 ≫ μ2, then T is stochastically invertible, i.e., there exists S: W → W such that S ○ T = IW ν1 a.s. and T ○ S = IW ν 2 a.s. If, in addition, ν1 = μ, then there exists a 1-convex function φ in the Gaussian Sobolev space double struck D sign 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.