Convergence rate of the occupation measure of classes of ergodic processes toward their invariant distribution in mean Wasserstein distance

TL;DR

This paper extends convergence rate bounds in Wasserstein distance for occupation measures of ergodic processes, including non-stationary and non-Markovian cases, with applications to diffusions and Gaussian processes.

Contribution

It generalizes existing convergence rate results to broader classes of ergodic processes without regularization, under conditional convergence assumptions.

Findings

Provides explicit convergence rate conditions for Brownian diffusions.

Extends bounds to processes driven by fractional Brownian motions.

Applies results to non-stationary and non-Markovian ergodic processes.

Abstract

N. Fournier and A. Guillin obtained in their 2015 PTRF paper some bounds of the L^p-mean rate of convergence in Wasserstein distance of empirical distributions for a class of stationary mixing processes. In this paper, we propose to extend their strategy of proof and provide general criterions which allow to keep similar rates for a larger class of processes. These results (which do not require regularization techniques) lead to various applications to occupation measures of ergodic processes which may be not stationary or not Markovian under an assumption of {\em conditional} convergence to equilibrium in Total Variation or Wasserstein distance. We then provide explicit conditions which lead to these rates for Brownian diffusions and additive SDEs driven by fractional Brownian Motions {or by Gaussian processes with stationary increments}.