Diffusions on Wasserstein Spaces

Abstract

We construct a canonical diffusion process on the space of probability measures over a closed Riemannian manifold, with invariant measure the Dirichlet–Ferguson measure. Together with a brief survey of the relevant literature, we collect several tools from the theory of point processes and of optimal transportation. Firstly, we study the characteristic functional of Dirichlet–Ferguson measures with non-negative finite intensity measure over locally compact Polish spaces. We compute such characteristic functional as a martingale limit of confluent Lauricella hypergeometric functions of type D with diverging arity. Secondly, we study the interplay between the self-conjugate prior property of Dirichlet distributions in Bayesian non-parametrics, the dynamical symmetry algebra of said Lauricella functions and Pólya Enumeration Theory. <br> Further, we provide a new proof of J. Sethuraman’s fixed point characterization of Dirichlet–Ferguson measures, and an understanding of the latter as an integral identity of Mecke- or Georgii–Nguyen–Zessin-type.<br> Thirdly, we prove a Rademacher-type result on the Wasserstein space over a closed Riemannian manifold. Namely, sufficient conditions are given for a probability measure P on the Wasserstein space, so that real-valued Lipschitz functions be P-a.e. differentiable in a suitable sense. Some examples of measures satisfying such conditions are also provided. Finally, we give two constructions of a Markov diffusion process with values in the said Wasserstein space. The process is associated with the Dirichlet integral induced by the Wasserstein gradient and by the Dirichlet–Ferguson measure with intensity the Riemannian volume measure of the base manifold. We study the properties of the process, including its invariant sets, short-time asymptotics for the heat kernel, and a description by means of a stochastic partial differential equation.