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2018 | OriginalPaper | Chapter

5. Criteria for Exponential Convergence to Quasi-Stationary Distributions and Applications to Multi-Dimensional Diffusions

Authors : Nicolas Champagnat, Koléhè Abdoulaye Coulibaly-Pasquier, Denis Villemonais

Published in: Séminaire de Probabilités XLIX

Publisher: Springer International Publishing

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Abstract

We consider general Markov processes with absorption and provide criteria ensuring the exponential convergence in total variation of the distribution of the process conditioned not to be absorbed. The first one is based on two-sided estimates on the transition kernel of the process and the second one on gradient estimates on its semigroup. We apply these criteria to multi-dimensional diffusion processes in bounded domains of \(\mathbb {R}^d\) or in compact Riemannian manifolds with boundary, with absorption at the boundary.

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The assumption of continuity is only used to ensure that the entrance times in compact sets are stopping times for the natural filtration (cf. e.g. [20, p. 48]), and hence that the strong Markov property applies at this time. Our result would also hold true for càdlàg (weak) Markov processes provided that the strong Markov property applies at the hitting times of compact sets.

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Title: Criteria for Exponential Convergence to Quasi-Stationary Distributions and Applications to Multi-Dimensional Diffusions
Authors: Nicolas Champagnat
Koléhè Abdoulaye Coulibaly-Pasquier
Denis Villemonais
Publisher: Springer International Publishing
Book: Séminaire de Probabilités XLIX
Print ISBN: 978-3-319-92419-9

Electronic ISBN: 978-3-319-92420-5

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-92420-5_5