Ornstein-Uhlenbeck Type Processes on Wasserstein Space

TL;DR

This paper constructs an Ornstein-Uhlenbeck process on the Wasserstein space of probability measures, establishing key functional inequalities and analyzing perturbations within this geometric stochastic framework.

Contribution

It introduces an OU-type Dirichlet form on Wasserstein space using stochastic analysis, providing new tools for infinite-dimensional probability geometry.

Findings

Log-Sobolev inequality holds for the constructed process.

The Markov semigroup is $L^2$-compact.

Perturbations of the Dirichlet form are analyzed.

Abstract

Let $\mathcal P_2$ be the space of probability measures on $\R^d$ 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.