Spatial birth and death processes as solutions of stochastic equations

TL;DR

This paper models spatial birth and death processes as solutions to stochastic equations, establishing conditions for their existence, uniqueness, and ergodic behavior, including exponential convergence to stationarity.

Contribution

It introduces a framework for solving spatial birth and death processes via stochastic equations, with new conditions for ergodicity and convergence in infinite population settings.

Findings

Existence and uniqueness conditions for solutions.

Ergodicity and exponential convergence under sub-critical birth rates.

Applicable to infinite populations over noncompact spaces.

Abstract

Spatial birth and death processes are obtained as solutions of a system of stochastic equations. The processes are required to be locally finite, but may involve an infinite population over the full (noncompact) type space. Conditions are given for existence and uniqueness of such solutions, and for temporal and spatial ergodicity. For birth and death processes with constant death rate, a sub-criticality condition on the birth rate implies that the process is ergodic and converges exponentially fast to the stationary distribution.