Measure transport on Wiener space and the Girsanov theorem

Abstract

Let ( W, H, μ) be an abstract Wiener space, and assume that ν i , i=1,2, are two probability measures on (W, B(W)) which are absolutely continuous with respect to μ. Assume that the Wasserstein distance between ν 1 and ν 2 is finite. Then there exists a map T= I W + ξ of W into itself such that ξ: W→ H is measurable and 1-cyclically monotone such that the image of ν 1 under T is ν 2. Moreover T is invertible on the support of ν 2. We give also some applications of this result such as the existence of the solutions of the Monge–Ampère equation in infinite dimensions. To cite this article: D. Feyel, A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1025–1028.