Unique quasi-stationary distribution, with a possibly stabilizing extinction

Aurélien Velleret

Stochastic Processes and their Applications, 2022, vol. 148, issue C, 98-138

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 conditioned 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.

Keywords: Quasi-stationary distribution; Survival capacity; Q-process; Harris recurrence; Birth-and-death process; Diffusion (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:148:y:2022:i:c:p:98-138

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DOI: 10.1016/j.spa.2022.02.004

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