Sobolev regularity for the Monge-Ampere equation in the Wiener space

TL;DR

This paper proves Sobolev regularity results for solutions to the Monge-Ampère equation in an infinite-dimensional Gaussian setting, establishing conditions for higher-order derivatives and change of variables formulas.

Contribution

It introduces Sobolev regularity results for the Monge-Ampère equation on Wiener space and provides conditions for the existence of third order derivatives.

Findings

Proves Sobolev regularity of the potential function  in optimal transport.

Establishes a change of variables formula involving the Carleman-Fredholm determinant.

Provides sufficient conditions for third order differentiability of .

Abstract

Given the standard Gaussian measure $\gamma$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g \cdot \gamma$ absolutely continuous with respect to $\gamma$, we consider the optimal transportation $T(x) = x + \nabla \varphi(x)$ of $g \cdot \gamma$ to $\gamma$. Assume that the function $|\nabla g|^2/g$ is $\gamma$-integrable. We prove that the function $\varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {\det}_2(I + D^2 \varphi) \exp \bigl(\mathcal{L} \varphi - 1/2 |\nabla \varphi|^2 \bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $\varphi$.