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.