The Evolution of a Spatial Stochastic Network

TL;DR

This paper analyzes the long-term behavior of a spatial stochastic network modeled by birth and death processes, demonstrating conditions for finite or infinite particle configurations and exploring invariant measures.

Contribution

It introduces new results on the convergence, uniqueness, and accumulation points of invariant measures in spatial birth-death stochastic networks.

Findings

Configurations converge in distribution under certain conditions.

Finite particle configurations occur when birth rate is less than death rate.

Examples show both finite and infinite limiting configurations.

Abstract

The asymptotic behavior of a stochastic network represented by a birth and death processes of particles on a compact state space is analyzed. Births: Particles are created at rate $\lambda_+$ and their location is independent of the current configuration. Deaths are due to negative particles arriving at rate $\lambda_-$. The death of a particle occurs when a negative particle arrives in its neighborhood and kills it. Several killing schemes are considered. The arriving locations of positive and negative particles are assumed to have the same distribution. By using a combination of monotonicity properties and invariance relations it is shown that the configurations of particles converge in distribution for several models. The problems of uniqueness of invariant measures and of the existence of accumulation points for the limiting configurations are also investigated. It is shown for several natural models that if $\lambda_+<\lambda_-$ then the asymptotic configuration has a finite number of points with probability 1. Examples with $\lambda_+<\lambda_-$ and an infinite number of particles in the limit are also presented.