Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes

Abstract

For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $(\mathbb{E}[{\mathbb{W}}_p^q({\mu }_T,\mu )]{)}^{\frac{1}{q}}(T\to \infty )$ uniformly in $(p,q)\in [1,\infty )\times (0,\infty )$, where ${\mu }_T$ is the empirical measure of the diffusion process, $\mu$ is the unique invariant probability measure, and ${\mathbb{W}}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that $\mathbb{E}|T{\mathbb{W}}_2^2({\mu }_T,\mu )-\Xi (T){|}^q\to 0$ as $T\to \infty$ for any $q\ge 1$, where $\Xi (T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.