We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension . 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 multidimensional base spaces to the modified massive Arratia flow over the unit interval described in V. Konarovskyi and M.-K. von Renesse (Comm. Pure Appl. Math. 72 (2019) 764–800). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.
Research supported by the Sonderforschungsbereich 1060 and the Hausdorff Center for Mathematics. The author gratefully acknowledges funding of his current position at IST Austria by the Austrian Science Fund (FWF) grant F65 and by the European Research Council (ERC, Grant agreement No. 716117, awarded to Prof. Dr. Jan Maas).
The author is especially grateful to Professors Karl-Theodor Sturm, Eugene Lytvynov, Maria Gordina, and Massimiliano Gubinelli for their several remarks and comments. He thanks Professor Günter Last for a preliminary version of  and Professors Federico Bassetti and Emanuele Dolera, and Dr. Carlo Orrieri, for discussions about the Dirichlet–Ferguson measure during his stay, in February 2018, at the Department of Mathematics “F. Casorati” of the Università degli Studi di Pavia. He thanks the Department for their kind hospitality. Finally, the author would like to express his gratitude to an anonymous reviewer whose remarks and comments improved the presentation of this work.
The entirety of this work was carried out while the author was a member of the Institute of Applied Mathematics of the University of Bonn.
"The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold." Ann. Probab. 50 (2) 591 - 648, March 2022. https://doi.org/10.1214/21-AOP1541