Sobolev regularity for the Monge-Ampere 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 $T(x)=x+\nabla \phi (x)$ 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 $g={\det }_2(I+D^2\phi )\exp (\mathcal{L}\phi -1/2|\nabla \phi {|}^2)$. We also establish sufficient conditions for the existence of third order derivatives of $\phi$.

22 pages. Some statements are corrected. More complete proofs are given

Cited by in corpus (3)

• Moment measures and stability for Gaussian inequalities
• Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry
• Backward Monge Potential and Monge-Ampere Equation