The Monge-Kantorovitch Problem and Monge-Ampere Equation on Wiener Space

Abstract

We give the solution of the Monge-Kantorovitch problem on the Wiener space for the singular Wasserstein metric which is defined with respect to the distance of the underlying Cameron-Martin space. We show, under the hypothesis that this distance is finite, the existence and the uniquness of the solutions, that they are supported by the graphs of the weak derivatives of H-convex Wiener functionals. then we prove the more general situation, where the measures are not even necessarily absolutely continuous w.r.to the Wiener measure. We give sufficient conditions for the hypothesis about the Wassestein distance is finite with the help of the Girsanov theorem. Finally we give the solutions of the Monge-Ampere equation using the classical Jacobi representation and/or the Ito parametrization of the Wiener space.