Sobolev regularity for the Monge–Ampère equation in the Wiener space

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 $\varphi$.