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.
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.
"Convergence to quasi-stationarity through Poincaré inequalities and Bakry-Émery criteria." Electron. J. Probab. 26 1 - 30, 2021. https://doi.org/10.1214/21-EJP644