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Regular version of the site

Article

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

Kyoto Journal of Mathematics. 2013. Vol. 53. No. 4. P. 713-738.

Numerous applications of the optimal transportation theory in finite-dimensional spaces have been found during the last decade. They include differential equations, probability theory, and geometry. The situation in infinite-dimensional spaces has been much less studied. However, some partial results on existence, uniqueness, and regularity have been obtained in recent papers of D. Feyel, A.S. Ustunel, M. Zakai, and the authors.

In this paper we study regularity properties of the infinite-dimensional transportation of measures on the Wiener space, where the transportation cost is given by the integral squared Cameron-Martin norm, the target measure is the Wiener measure, and the source measure is absolutely continuous with respect to the Wiener measure. Assuming that the density has the finite Fisher information (belongs to a certain Sobolev class), we prove that the potential of the corresponding optimal transportation belongs to the second (weighted) Sobolev space W^{2,2}. Some estimates involving higher-order derivatives are given. This result settles a long-standing problem and is of principal importance for the whole area of infinite-dimensional optimal transport and its applications in stochastic analysis and measure theory in infinite dimensions.