Quasi-stationary distributions for discrete-state models

This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth–death processes and related models. The question of under what circumstances a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, is addressed as well. We conclude with a discussion of computational aspects, with more details given in a web appendix accompanying this paper.

► The paper contains a survey of results related to quasi-stationary distributions.
► Our main concern is with continuous-time Markov chains over a finite or countably infinite state space.
► We discuss the import of quasi-stationary distributions as descriptors of long-term behaviour.
► We review results on the existence and identification of quasi-stationary distributions.
► We give a summary of recent developments in numerical and approximation methods.