Asymptotics in Wasserstein Distance for Empirical Measures of Markov Processes

TL;DR

This paper investigates the convergence rates of empirical measures of Markov processes in Wasserstein distance, providing sharp bounds and techniques applicable to diffusion processes on manifolds and general ergodic Markov processes.

Contribution

It offers new explicit convergence rate estimates for empirical measures of Markov processes, including sharp bounds for diffusion processes on manifolds and methods for general ergodic processes.

Findings

Sharp convergence rates for diffusion processes on manifolds.

Explicit estimates for ergodic Markov processes.

Techniques for estimating Wasserstein distance of empirical measures.

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.