Subgeometric rates of convergence of Markov processes in the Wasserstein metric

TL;DR

This paper proves subgeometric convergence rates for Markov processes in Wasserstein metric, linking Lyapunov conditions and small sets to convergence speed, with applications to stochastic delay differential equations.

Contribution

It establishes new subgeometric convergence bounds for Markov processes in Wasserstein metric under Lyapunov and small set conditions, extending to continuous time and stochastic delay equations.

Findings

Lyapunov drift and small set conditions imply subgeometric convergence.

Continuous time results require supermartingale conditions.

Veretennikov-Khasminskii condition ensures subexponential convergence.

Abstract

We establish subgeometric bounds on convergence rate of general Markov processes in the Wasserstein metric. In the discrete time setting we prove that the Lyapunov drift condition and the existence of a "good" $d$-small set imply subgeometric convergence to the invariant measure. In the continuous time setting we obtain the same convergence rate provided that there exists a "good" $d$-small set and the Douc-Fort-Guillin supermartingale condition holds. As an application of our results, we prove that the Veretennikov-Khasminskii condition is sufficient for subexponential convergence of strong solutions of stochastic delay differential equations.