# Variational characterization of the regularity of Monge-Brenier maps

## TL;DR

This paper investigates the regularity of Monge-Brenier maps on Wiener spaces, demonstrating their Sobolev regularity through variational methods and large deviations theory, under specific measure assumptions.

## Contribution

It introduces a variational approach to establish Sobolev regularity of Monge-Brenier maps in an abstract Wiener space setting, extending previous results.

## Key findings

- Monge-Brenier maps are Sobolev regular under finite information hypothesis.
- Variational methods can be used to analyze the regularity of optimal transport maps.
- Results apply to both forward and backward Monge-Brenier maps.

## Abstract

On an abstract Wiener space, assume that T is the solution of the quadratic Monge problem associated to the Wiener measure and a second one with a Radon-Nikodym derivative of exponential type. Under the finite information hypothesis, using a variational method, we prove that T minimizes a certain functional originating from the large deviations theory. Applying a variational method a la Euler, we obtain the Sobolev regularity of the backward Monge-Brenier map. A similar result also holds for the forward Monge-Brenier map.