Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction

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.

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Cited by in corpus (4)

• A non-conservative Harris ergodic theorem
• Two level natural selection with a quasi-stationarity approach
• A central limit theorem for continuous-time Markov processes conditioned not to be absorbed
• A note on the jump locations of Markov processes