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

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