Convergence rates for the $p$-Wasserstein distance of the empirical measures of an ergodic Markov process

TL;DR

This paper establishes upper bounds on the convergence rate of empirical measures to the invariant measure in the $p$-Wasserstein distance for ergodic Markov processes, including diffusions and Langevin dynamics.

Contribution

It provides new convergence rate bounds for empirical measures of ergodic Markov processes under various contractivity and moment conditions.

Findings

Derived upper bounds for Wasserstein distances in ergodic Markov processes.

Applicable to diffusions and Langevin dynamics.

Results depend on contractivity and moment conditions.

Abstract

Let $X:=(X_t)_{t\geq 0}$ be an ergodic Markov process on $\real^d$, and $p>0$. We derive upper bounds of the $p$-Wasserstein distance between the invariant measure and the empirical measures of the Markov process $X$. For this we assume, e.g.\ that the transition semigroup of $X$ is exponentially contractive in terms of the $1$-Wasserstein distance, or that the iterated Poincar\'e inequality holds together with certain moment conditions on the invariant measure. Typical examples include diffusions and underdamped Langevin dynamics.