Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction

TL;DR

This paper provides new conditions for exponential convergence to a unique quasi-stationary distribution in Markov processes, incorporating initial condition dependence, with applications to birth-death and diffusion processes.

Contribution

It introduces a novel coupling technique that depends on both survival horizon and initial distribution, advancing the understanding of quasi-stationary distributions.

Findings

Established exponential convergence conditions for QSDs.

Proved existence and ergodicity of the Q-process.

Applied results to birth-death and diffusion models.

Abstract

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditionned upon never being absorbed. The technique relies on a coupling procedure that is related to Harris recurrence (for Markov Chains). It applies to general continuous-time and continuous-space Markov processes. The main novelty is that we modulate each coupling step depending both on a final horizon of time (for survival) and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment.