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

Variational characterization of the regularity of Monge-Brenier maps