Regularity of the backward Monge potential and the Monge–Ampère equation on Wiener space

Abstract

. In this paper, the Monge–Kantorovich problem is considered in infinite dimensions on an abstract Wiener space ( W, H, µ ) , where H is the Cameron–Martin space and µ 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 µ . 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–Ampère equation.