# The Dirichlet--Ferguson Diffusion on the space of probability measures over a closed Riemannian manifold

## Abstract

We construct a diffusion process on the $L^2$-Wasserstein space $P_2(M)$ over a closed Riemannian manifold $M$. The process, which may be regarded as a candidate for the Brownian motion on $P_2(M)$, is associated with the Dirichlet form induced by the $L^2$-Wasserstein gradient and by the Dirichlet--Ferguson random measure with intensity the Riemannian volume measure on $M$. We discuss the closability of the form via an integration-by-parts formula, which allows explicit computations for the generator and a specification of the process via a measure-valued martingale problem. We comment how the construction is related to previous work of von Renesse--Sturm on the Wasserstein Diffusion, of Kondratiev--Lytvynov--Vershik on diffusions on the cone of Radon measures, and of Konarovskyi--von Renesse on the Modified Massive Arratia Flow. Ann. Probab. 50(2):591–648, 2022.

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