Solution of the Monge–Ampère equation onWiener space for general log-concave measures

In this work we prove that the unique 1-convex solution of the Monge–Kantorovitch measure transportation problem between the Wiener measure and a target measure which has an Hlog- concave density, in the sense of Feyel and Üstünel [J. Funct. Anal. 176 (2000) 400–428], w.r.t the Wiener measure is also the strong solution of the Monge–Ampère equation in the frame of infinite-dimensional Fréchet spaces. We further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of Monge–Kantorovitch and clarify the relation between this concept and the Itô-solutions of the Monge–Ampère equation. © 2005 Elsevier Inc. All rights reserved.