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

Abstract

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