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

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 $(\mathbb{E}[{\mathbb{W}}_p^q({\mu }_T,\mu )]{)}^{\frac{1}{q}}(T\to \mathrm{\infty })$ uniformly in $(p,q)\in [1,\mathrm{\infty })\times (0,\mathrm{\infty })$, 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 $\mathbb{E}\mathrm{\mid }T{\mathbb{W}}_2^2({\mu }_T,\mu )-\mathrm{\Xi }(T){\mathrm{\mid }}^q\to 0$ as $T\to \mathrm{\infty }$ for any $q\ge 1$, where $\mathrm{\Xi }(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.