Exponential convergence to quasi-stationary distribution and Q-process

Nicolas Champagnat 1, 2 Denis Villemonais 1, 2
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : For general, almost surely absorbed Markov processes, we obtain necessary and 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 to never be absorbed). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinite-dimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.
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Nicolas Champagnat, Denis Villemonais. Exponential convergence to quasi-stationary distribution and Q-process. Probability Theory and Related Fields, Springer Verlag, 2016, 164 (1), pp.243-283. ⟨http://link.springer.com/article/10.1007%2Fs00440-014-0611-7#⟩. ⟨10.1007/s00440-014-0611-7⟩. ⟨hal-00973509v2⟩

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