Variational characterization of the regularity of Monge-Brenier maps
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.
arXiv admin note: text overlap with arXiv:math/0403497, arXiv:math/0407106