Uniform Wasserstein convergence of penalized Markov processes

TL;DR

This paper introduces a simple criterion for uniform Wasserstein convergence of penalized Markov processes with soft killing, applicable to complex systems like Bernoulli convolutions and switched dynamical systems.

Contribution

It provides a new criterion ensuring convergence in Wasserstein distance for a broad class of penalized Markov processes, including cases where total variation convergence fails.

Findings

Criterion successfully applied to Bernoulli convolutions

Criterion applicable to piecewise deterministic Markov processes

Convergence in Wasserstein distance established where total variation fails

Abstract

For general penalized Markov processes with soft killing, we propose a simple criterion ensuring uniform convergence of conditional distributions in Wasserstein distance to a unique quasi-stationary distribution. We give several examples of application where our criterion can be checked, including Bernoulli convolutions and piecewise deterministic Markov processes of the form of switched dynamical systems, for which convergence in total variation is not possible.