Wasserstein space over the Wiener space

The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel–Üstünel. Carrying out the program of Ambrosio–Gigli–Savaré, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein–Uhlenbeck evolution equation.