Ornstein[formula omitted]Uhlenbeck type processes on Wasserstein spaces

Abstract

Let P2 be the space of probability measures on Rd having finite second moment, and consider the Riemannian structure on P2 induced by the intrinsic derivative on the L2-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein−Uhlenbeck (OU) type Dirichlet form on P2 whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is L2-compact. Perturbations of the OU Dirichlet form are also studied.