{"paper":{"title":"Sobolev regularity for the Monge-Ampere equation in the Wiener space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alexander V. Kolesnikov, Vladimir I. Bogachev","submitted_at":"2011-10-09T11:40:38Z","abstract_excerpt":"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 $T(x) = x + \\nabla \\varphi(x)$ of $g \\cdot \\gamma$ to $\\gamma$. Assume that the function $|\\nabla g|^2/g$ is $\\gamma$-integrable. We prove that the function $\\varphi$ 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1822","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}