Ornstein-Uhlenbeck Type Processes on Wasserstein Space

Abstract

Let ${\mathcal{P}}_2$ be the space of probability measures on ${\mathbb{R}}^d$ having finite second moment, and consider the Riemannian structure on ${\mathcal{P}}_2$ induced by the intrinsic derivative on the $L^2$-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein$-$Uhlenbeck (OU) type Dirichlet form on ${\mathcal{P}}_2$ whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is $L^2$-compact. Perturbations of the OU Dirichlet form are also studied.

arXiv comment28 pages