Semigroup approach to birth-and-death stochastic dynamics in continuum

TL;DR

This paper develops a semigroup-based framework for modeling birth-and-death stochastic processes in continuous space, establishing conditions for their existence and analyzing their scaling limits.

Contribution

It introduces a general semigroup approach for birth-and-death dynamics in continuum and provides conditions ensuring well-defined evolution and scaling convergence.

Findings

Established conditions for the existence of a strongly continuous semigroup.

Proved convergence of a Vlasov-type scaling for the dynamics.

Provided a framework for correlation functions satisfying the Ruelle bound.

Abstract

We describe a general approach to the construction of a state evolution corresponding to the Markov generator of a spatial birth-and-death dynamics in $\mathbb{R}^d$. We present conditions on the birth-and-death intensities which are sufficient for the existence of an evolution as a strongly continuous semigroup in a proper Banach space of correlation functions satisfying the Ruelle bound. The convergence of a Vlasov-type scaling for the corresponding stochastic dynamics is considered.