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We construct a recurrent diffusion process with values in the space ofprobability measures over an arbitrary closed Riemannian manifold of dimension$d\ge 2$. The process is associated with the Dirichlet form defined byintegration of the Wasserstein gradient w.r.t. the Dirichlet-Ferguson measure,and is the counterpart on multi-dimensional base spaces to the Modified MassiveArratia Flow over the unit interval described in V. Konarovskyi, M.-K. vonRenesse, Comm. Pure Appl. Math., 72, 0764-0800 (2019). Together with twodifferent constructions of the process, we discuss its ergodicity, invariantsets, finite-dimensional approximations, and Varadhan short-time asymptotics.

The Dirichlet-Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold