Lattice birth-and-death processes

TL;DR

This paper constructs and analyzes lattice birth-and-death processes, establishing their fundamental properties and conditions for invariant distributions, with applications to ecological and biological models.

Contribution

It provides a rigorous construction and characterization of lattice birth-and-death processes, including existence, uniqueness, and invariant measures, with applications to specific ecological models.

Findings

Established pathwise uniqueness and strong existence of the processes.

Provided conditions for invariant distributions using Lyapunov functions.

Applied results to ecological models like the Bolker--Pacala--Dieckmann--Law model.

Abstract

Lattice birth-and-death Markov dynamics of particle systems with spins from the set of non-negative integers are constructed as unique solutions to certain stochastic equations. Pathwise uniqueness, strong existence, Markov property and joint uniqueness in law are proven, and a martingale characterization of the process is given. Sufficient conditions for the existence of an invariant distribution are formulated in terms of Lyapunov functions. We apply obtained results to discrete analogs of the Bolker--Pacala--Dieckmann--Law model and an aggregation model.