Invariant measure

Dirichlet distribution

Pure mathematics

Probability measure

Dirichlet form

Mathematics

Manifold (fluid mechanics)

Knowledge management

Dirichlet's energy

Ergodic theory

Statistics

Engineering

Mathematical analysis

Philosophy

Boundary value problem

Ergodicity

Space (punctuation)

Riemannian manifold

Measure (data warehouse)

Computer science

Linguistics

Innovation diffusion

Diffusion process

Database

Mechanical engineering

The Annals of Probability

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.

The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold