Regularity of the backward monge potential and the monge-ampere equation on wiener space

Abstract

In this paper, the Monge-Kantorovich problem is considered in infinite dimensions on an abstract Wiener space (W, H, mu), where H is the Cameron-Martin space and mu is the Gaussian measure. We study the regularity of optimal transport maps with a quadratic cost function assuming that both initial and target measures have a strictly positive Radon-Nikodym density with respect to mu. Under some conditions on the density functions, the forward and backward transport maps can be written in terms of Sobolev derivatives of so-called Monge-Brenier maps, or Monge potentials. We show the Sobolev regularity of the backward potential under the assumption that the density of the initial measure is log-concave and prove that the backward potential solves the Monge-Ampere equation.