Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates

TL;DR

This paper investigates the long-term behavior of multi-dimensional birth and death processes conditioned on survival, introducing new Lyapunov function techniques to handle unbounded absorption rates and proving exponential convergence to a unique quasi-stationary distribution.

Contribution

It develops novel Lyapunov function methods for unbounded killing rates and establishes exponential convergence of the conditioned process to a unique distribution.

Findings

Exponential convergence in total variation to a unique quasi-stationary distribution.

Applicable to models with unbounded absorption rates and various competition structures.

Provides explicit conditions for neutral competition cases.

Abstract

This article studies the quasi-stationary behaviour of multidimensional birth and death processes, modeling the interaction between several species, absorbed when one of the coordinates hits 0. We study models where the absorption rate is not uniformly bounded, contrary to most of the previous works. To handle this natural situation, we develop original Lyapunov function arguments that might apply in other situations with unbounded killing rates. We obtain the exponential convergence in total variation of the conditional distributions to a unique stationary distribution, uniformly with respect to the initial distribution. Our results cover general birth and death models with stronger intra-specific than inter-specific competition, and cases with neutral competition with explicit conditions on the dimension of the process.