A Hard-core Stochastic Process with Simultaneous Births and Deaths

TL;DR

This paper introduces a stochastic spatial point process with simultaneous births and deaths that maintains a hard-core property, providing explicit construction, stability analysis, and convergence results.

Contribution

It presents a new explicit construction of a hard-core stochastic process with births and deaths, and proves its stochastic stability and exponential convergence to stationarity.

Findings

Explicit construction of the process

Proof of stochastic stability and stationarity

Exponential convergence to the stationary distribution

Abstract

We consider a stochastic spatial point process with births and deaths on $\mathbb{R}^d$, with the hard-core property that at any time the balls of radius half of any two points do not overlap. We give explicit construction of the process. Under some more conditions, we show the stochastic stability of this Markov process by constructing a stationary regime for the dynamics. The main tool used to construct the stationary regime is the coupling form the past technique. Further, we study the exponential convergence of the probability distribution to the stationary distribution.