Existence of a Unique Quasi-stationary Distribution for Stochastic Reaction Networks

TL;DR

This paper establishes conditions for the existence and uniqueness of quasi-stationary distributions in stochastic reaction networks, providing insights into their long-term behavior before extinction.

Contribution

It introduces the concepts of absorbing and endorsed sets and proves the existence and uniqueness of quasi-stationary distributions within these sets for reaction networks.

Findings

Existence of a globally attracting quasi-stationary distribution under certain conditions.

Exponential convergence to the quasi-stationary distribution from any initial distribution.

Sufficient conditions for the existence and uniqueness of quasi-stationary distributions.

Abstract

In the setting of stochastic dynamical systems that eventually go extinct, the quasi-stationary distributions are useful to understand the long-term behavior of a system before evanescence. For a broad class of applicable continuous-time Markov processes on countably infinite state spaces, known as reaction networks, we introduce the inferred notion of absorbing and endorsed sets, and obtain sufficient conditions for the existence and uniqueness of a quasi-stationary distribution within each such endorsed set. In particular, we obtain sufficient conditions for the existence of a globally attracting quasi-stationary distribution in the space of probability measures on the set of endorsed states. Furthermore, under these conditions, the convergence from any initial distribution to the quasi-stationary distribution is exponential in the total variation norm.