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Unique quasi-stationary distribution, with a possibly stabilizing extinction

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  • Velleret, Aurélien

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.

Suggested Citation

  • Velleret, Aurélien, 2022. "Unique quasi-stationary distribution, with a possibly stabilizing extinction," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 98-138.
    • Handle: RePEc:eee:spapps:v:148:y:2022:i:c:p:98-138
    DOI: 10.1016/j.spa.2022.02.004
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    References listed on IDEAS

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    1. Champagnat, Nicolas & Villemonais, Denis, 2021. "Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 51-74.
    2. van Doorn, Erik A. & Pollett, Philip K., 2013. "Quasi-stationary distributions for discrete-state models," European Journal of Operational Research, Elsevier, vol. 230(1), pages 1-14.
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