Exponential convergence to quasi-stationary distribution and Q-process

TL;DR

This paper establishes necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in absorbed Markov processes, and demonstrates their application across various stochastic models.

Contribution

It provides a comprehensive set of criteria for exponential convergence and ergodicity of the Q-process in general absorbed Markov processes, extending to multiple complex models.

Findings

Conditions for exponential convergence are characterized.

Existence and ergodicity of the Q-process are proven.

Applications include birth-death processes, population models, and neutron transport.

Abstract

For general, almost surely absorbed Markov processes, we obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the $Q$-process (the process conditioned to never be absorbed). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinite-dimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.