Wasserstein Convergence Rate for Empirical Measures of Markov Processes

TL;DR

This paper establishes the convergence rate of empirical measures in Wasserstein distance for ergodic Markov processes, providing sharp estimates and extending applicability to complex models like Hamiltonian systems and jump processes.

Contribution

It offers new sharp convergence rate estimates for empirical measures of ergodic Markov processes, including models previously not covered by existing results.

Findings

Wasserstein convergence rates are sharp in certain cases.

Application to models like Hamiltonian systems and jump processes.

Extension of convergence results to a broader class of Markov processes.

Abstract

The convergence rate in Wasserstein distance is estimated for empirical measures of ergodic Markov processes, and the estimate can be sharp in some specific situations. The main result is applied to subordinations of typical models excluded by existing results, which include: stochastic Hamiltonian systems on $\mathbb R^{n}\times \mathbb R^{m}$, spherical velocity Langevin processes on $\mathbb R^n\times\mathbb S^{n-1},$ multi-dimensional Wright-Fisher type diffusion processes, and stable type jump processes.