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