Backward Monge Potential and Monge-Ampere Equation

Abstract

In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space $(W, H,\mu)$, where $H$ is 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 conditions on the density functions, the forward and backward transport maps can be written in terms of Sobolev derivative of so-called Monge-Brenier maps, or Monge potentials. We show Sobolev regularity of the backward potential under the assumption that the density of the initial measure is log-concave and prove that it solves Monge-Ampere equation.