Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
Feng-Yu Wang
Tianjin University, China; Swansea University, UK
![Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-jems-volume-25-issue-9.png&w=3840&q=90)
Abstract
Let be a -dimensional connected compact Riemannian manifold with boundary , let be such that is a probability measure, and let be the diffusion process generated by with . Consider the empirical measure under the condition for the diffusion process. If , then for any initial distribution not fully supported on ,
holds for some constant with when is convex, where for the first Dirichet eigenfunction of , are the Dirichlet eigenvalues of listed in increasing order counting multiplicities, and the upper bound is finite if and only if . When , decays on the order of , while for it behaves like , as .
Cite this article
Feng-Yu Wang, Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds. J. Eur. Math. Soc. 25 (2023), no. 9, pp. 3695–3725
DOI 10.4171/JEMS/1269