Open Access
2023 General criteria for the study of quasi-stationarity
Nicolas Champagnat, Denis Villemonais
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Electron. J. Probab. 28: 1-84 (2023). DOI: 10.1214/22-EJP880


For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.

Funding Statement

This work was partially funded by the Chair “Modélisation Mathématique et Biodiversité” of VEOLIA-Ecole Polytechnique-MNHN-F.X. N.C. was partially funded by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.


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Nicolas Champagnat. Denis Villemonais. "General criteria for the study of quasi-stationarity." Electron. J. Probab. 28 1 - 84, 2023.


Received: 23 November 2018; Accepted: 8 November 2022; Published: 2023
First available in Project Euclid: 8 February 2023

MathSciNet: MR4546021
zbMATH: 07707081
Digital Object Identifier: 10.1214/22-EJP880

Primary: 37A25 , 60B10 , 60F99 , 60J05 , 60J10 , 60J25 , 60J27
Secondary: 60J60 , 60J75 , 60J80 , 93E03

Keywords: birth and death processes , Diffusion processes , Galton-Watson processes , Markov processes with absorption , mixing property , perturbed dynamical systems , Q-process , quasi-stationary distribution , reducible processes

Vol.28 • 2023
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