Measure transport on Wiener space and the Girsanov theorem. (Q1608731)

Let \((W, H, \mu)\) be an abstract Wiener space. Assume that \(\nu_1, \nu_2\) are two probability measures on \((W, \mathcal{B}(W))\) absolutely continuous with respect to \(\mu.\) If the Wasserstein distance \(d_W(\nu_1, \nu_2)\) is finite, then it is proved that there exists a map \(T=I_W+\xi: W\to W\) such that \(\nu_2=T\nu_1,\) where \(\xi\) maps \(W\) into \(H\), is measurable and \(1\)-cyclically monotone. Moreover, \(T\) is invertible on the support of \(\nu_2\). Some applications of this result are given to the Monge-Ampere equation in infinite-dimensional spaces.