On Spatial Point Processes with Uniform Births and Deaths by Random Connection

TL;DR

This paper studies a class of spatial birth-death processes in Euclidean space where births are uniform and deaths depend on a shot noise function, establishing their stationary regime and point repulsion properties.

Contribution

It constructs the unique stationary regime for these processes and introduces the concept of $f$-repulsion between points.

Findings

Established the existence and uniqueness of the stationary regime.

Derived balance integral relations for factorial moment measures.

Proved the process exhibits $f$-repulsion between points.

Abstract

This paper is focused on a class of spatial birth and death process of the Euclidean space where the birth rate is constant and the death rate of a given point is the shot noise created at its location by the other points of the current configuration for some response function $f$. An equivalent view point is that each pair of points of the configuration establishes a random connection at an exponential time determined by $f$, which results in the death of one of the two points. We concentrate on space-motion invariant processes of this type. Under some natural conditions on $f$, we construct the unique time-stationary regime of this class of point processes by a coupling argument. We then use the birth and death structure to establish a hierarchy of balance integral relations between the factorial moment measures. Finally, we show that the time-stationary point process exhibits a certain kind of repulsion between its points that we call $f$-repulsion.