Variational characterization of the regularity of Monge-Brenier maps

2017/04/02 by Ustunel, Ali Suleyman
#35Q #60H #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1704.00310

Abstract

On an abstract Wiener space, assume that T is the solution of the quadratic Monge problem associated to the Wiener measure and a second one with a Radon-Nikodym derivative of exponential type. Under the finite information hypothesis, using a variational method, we prove that T minimizes a certain functional originating from the large deviations theory. Applying a variational method a la Euler, we obtain the Sobolev regularity of the backward Monge-Brenier map. A similar result also holds for the forward Monge-Brenier map.

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