Uniform Wasserstein convergence of penalized Markov processes
Nicolas Champagnat, Edouard Strickler, Denis Villemonais
TL;DR
The paper develops a general criterion for uniform exponential convergence in Wasserstein distance of conditional distributions of penalized Markov processes with soft killing to a unique quasi-stationary distribution. It centers on an exponential penalized coupling assumption (A) that controls the distance between two coupled unkilled trajectories under the penalization, yielding a contraction that is uniform in the initial condition and implies existence and uniqueness of the quasi-stationary distribution, as well as ergodic properties of the Q-process and quasi-ergodic results. The authors provide explicit formulations for contraction rates, extend the framework to subexponential couplings, and apply the theory to Bernoulli convolutions and switched dynamical systems, including PDMPs, where total variation convergence fails but Wasserstein convergence holds. They also derive practical criteria linking the coupling rate of the non-penalized process to the killing rate, discuss almost sure contraction and iterated random contracting maps, and present a counterexample demonstrating the necessity of Lipschitz regularity on the killing rate for the main results. Overall, the work broadens the reach of Wasserstein-based convergence analysis for absorbed processes and offers tangible tools for establishing quasi-stationarity and long-time behavior in systems where traditional total variation methods fail.
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.