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

L. Dello Schiavo

doi:10.1214/21-aop1541

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arXiv · 1811.11598

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

L. Dello Schiavo

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Abstract

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

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BibTeX RIS

L. Dello Schiavo. 2018-11-28. The Dirichlet-Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold. https://doi.org/10.1214/21-aop1541

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