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

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.

51 pages, 2 figures, part of the appendix is now arXiv:2003.01366

References in corpus (5)

• A Rademacher-type Theorem on $L^2$-Wasserstein Spaces over Closed Riemannian Manifolds
• Geometric properties of Dirichlet forms under order isomorphisms
• Characteristic Functionals of Dirichlet Measures
• A Mecke-type characterization of the Dirichlet-Ferguson measure
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Cited by in corpus (11)

• On Dean-Kawasaki Dynamics with Smooth Drift Potential
• A Rademacher-type Theorem on $L^2$-Wasserstein Spaces over Closed Riemannian Manifolds
• Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces
• Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry
• A Mecke-type characterization of the Dirichlet-Ferguson measure
• Ergodic Decompositions of Dirichlet Forms under Order Isomorphisms
• Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications
• Dean-Kawasaki equation with initial condition in the space of positive distributions
• Persistence of Rademacher-type and Sobolev-to-Lipschitz properties
• Ill-posedness of the pure-noise Dean-Kawasaki equation
• Weak Error of Dean-Kawasaki Equation with Smooth Mean-Field Interactions