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

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

2011-10-09

arxiv.org/abs/1110.1822v1

Mathematics

Functional Analysis

14 pages

Scientific paper

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 Sobolev regular and satisfies the classical change of variables formula $g = {\det}_2(I + D^2 \phi) \exp \bigl(\mathcal{L} \phi - 1/2 |\nabla \phi|^2 \bigr)$. We also establish sufficient conditions for the existence of third-order derivatives of $\phi$.

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