Asymptotics in Wasserstein Distance for Empirical Measures of Markov Processes

Abstract: In this paper we introduce some recent progresses on the convergence rate in Wasserstein distance for empirical measures of Markov processes. For diffusion processes on compact manifolds possibly with reflecting or killing boundary conditions, the sharp convergence rate as well as renormalization limits are presented in terms of the dimension of the manifold and the spectrum of the generator. For general ergodic Markov processes, explicit estimates are presented for the convergence rate by using a nice reference diffusion process, which are illustrated by some typical examples. Finally, some techniques are introduced to estimate the Wasserstein distance of empirical measures.