Existence and uniqueness of a quasi-stationary distribution for Markov processes with fast return from infinity

TL;DR

This paper proves the existence, uniqueness, and exponential convergence to a quasi-stationary distribution for certain Markov processes on countable spaces, especially those with rapid return from infinity, with applications to birth-death processes.

Contribution

It establishes conditions for the existence and uniqueness of quasi-stationary distributions and provides explicit convergence rates, including for non-irreducible processes.

Findings

Unique quasi-stationary distribution exists under rapid return conditions.

Processes converge exponentially fast to the quasi-stationary distribution.

Results apply to non-irreducible processes on countable spaces.

Abstract

We study the long time behaviour of a Markov process evolving in $\mathbb{N}$ and conditioned not to hit 0. Assuming that the process comes back quickly from infinity, we prove that the process admits a unique quasi-stationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to infinity). Moreover, we prove that the distribution of the process converges exponentially fast in total variation norm to its quasi-stationary distribution and we provide an explicit rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasi-stationary distribution for birth and death processes: we prove that these processes converge exponentially fast to a quasi-stationary distribution if and only if they have a unique quasi-stationary distribution. Also, considering the lack of results on quasi-stationary distributions for non-irreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasi-stationary distribution for a class of possibly non-irreducible processes.