Backward Monge Potential and Monge-Ampere Equation

TL;DR

This paper investigates the regularity of optimal transport maps in infinite-dimensional Gaussian spaces, establishing Sobolev regularity of backward potentials and their relation to the Monge-Ampere equation under certain density conditions.

Contribution

It introduces conditions under which the backward Monge potential is Sobolev regular and solves the Monge-Ampere equation in an infinite-dimensional setting.

Findings

Backward potential is Sobolev regular under log-concavity.

Backward potential solves the Monge-Ampere equation.

Results extend optimal transport theory to infinite-dimensional spaces.

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.