Sharp Lq-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 ( E [ Wₚq(μT,μ)])1/q (T → ∞) uniformly in (p,q)∈ [1,∞) × (0,∞), where μT is the empirical measure of the diffusion process, μ is the unique invariant probability measure, and Wₚ is the p-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that E |T W₂²(μT,μ)-Ξ(T)|q → 0 as T→∞ for any q≥ 1, where Ξ(T) is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.