# Subexponential upper and lower bounds in Wasserstein distance for Markov processes

## TL;DR

This paper establishes subexponential and exponential convergence bounds in Wasserstein distance for various Markov processes, providing sharp characterizations of convergence rates under different conditions.

## Contribution

It introduces new subexponential bounds for Wasserstein convergence in Markov processes using Foster-Lyapunov conditions and applies these to specific stochastic models.

## Key findings

- Subexponential convergence bounds for irreducible, aperiodic Markov processes.
- Exponential ergodicity under asymptotic flatness for Itô processes.
- Sharp rate characterizations for Langevin, Ornstein-Uhlenbeck, and recurrence time chains.

## Abstract

In this article, relying on Foster-Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the $\mathrm{L}^p$-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov L\'evy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the $\mathrm{L}^p$-Wasserstein distance for a class of It\^{o} processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes are presented, including Langevin tempered diffusion processes, piecewise Ornstein-Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we provide a sharp characterization of the rate of convergence via matching upper and lower bounds.