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2021 Convergence to quasi-stationarity through Poincaré inequalities and Bakry-Émery criteria
William Oçafrain
Author Affiliations +
Electron. J. Probab. 26: 1-30 (2021). DOI: 10.1214/21-EJP644


This paper aims to provide some tools coming from functional inequalities to deal with quasi-stationarity for absorbed Markov processes. First, it is shown how a Poincaré inequality related to a suitable Doob transform entails exponential convergence of conditioned distributions to a quasi-stationary distribution in total variation and in 1-Wasserstein distance. A special attention is paid to multi-dimensional diffusion processes, for which the aforementioned Poincaré inequality is implied by an easier-to-check Bakry-Émery condition depending on the right eigenvector for the sub-Markovian generator, which is not always known. Under additional assumptions on the potential, it is possible to bypass this lack of knowledge showing that exponential quasi-ergodicity is entailed by the classical Bakry-Émery condition.

Funding Statement

This research was supported by the Swiss National Foundation grant 200020 196999.


I am very grateful to the anonymous referee for his/her comments and questions, which allow me to better the paper.


Download Citation

William Oçafrain. "Convergence to quasi-stationarity through Poincaré inequalities and Bakry-Émery criteria." Electron. J. Probab. 26 1 - 30, 2021.


Received: 27 January 2020; Accepted: 4 May 2021; Published: 2021
First available in Project Euclid: 16 June 2021

Digital Object Identifier: 10.1214/21-EJP644

Primary: ‎39B62 , 60B10 , 60F99 , 60J25 , 60J50 , 60J60.

Keywords: 1-Wasserstein distance , Absorbed Markov processes , Bakry-Émery condition , multi-dimensional diffusion processes , Poincaré inequality , quasi-stationary distribution

Vol.26 • 2021
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