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 $T\left(x\right)=x+\nabla \phi \left(x\right)$ of $g\cdot \gamma$ to $\gamma$. Assume that the function $|\nabla g{|}^2/g$ is $\gamma$-integrable. We prove that the function $\phi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g={det}_2(\mathrm{I}+D^2\phi )exp(\mathcal{L}\phi -\frac{1}{2}|\nabla \phi {|}^2)$. We also establish sufficient conditions for the existence of third-order derivatives of $\phi$.