Persistence of competing systems of branching random walks

TL;DR

This paper studies the long-term behavior of systems of branching random walks on the real line, identifying conditions under which the system persists or goes extinct, and characterizing the invariant point processes.

Contribution

It provides a precise criterion based on the log-Laplace transform for the persistence or extinction of branching random walk systems and characterizes their invariant point processes.

Findings

Persistence occurs when $ ext{sign}(\lambda ext{phi'}(\lambda))$ is positive.

Extinction occurs when $ ext{sign}(\lambda ext{phi'}(\lambda))$ is negative.

Invariant point processes are mixtures of scaled cluster-invariant processes.

Abstract

We consider a system of independent branching random walks on $\R$ which start off a Poisson point process with intensity of the form $e_{\lambda}(du)=e^{-\lambda u}du$, where $\lambda\in\R$ is chosen in such a way that the overall intensity of particles is preserved. Denote by $\chi$ the cluster distribution and let $\phi$ be the log-Laplace transform of the intensity of $\chi$. If $\lambda\phi'(\lambda)>0$, we show that the system is persistent (stable) meaning that the point process formed by the particles in the $n$-th generation converges as $n\to\infty$ to a non-trivial point process $\Pi_{e_{\lambda}}^{\chi}$ with intensity $e_{\lambda}$. If $\lambda\phi'(\lambda)<0$, then the branching population suffers local extinction meaning that the limiting point process is empty. We characterize (generally, non-stationary) point processes on $\R$ which are cluster-invariant with respect to the cluster distribution $\chi$ as mixtures of the point processes $\Pi_{ce_{\lambda}}^{\chi}$ over $c>0$ and $\lambda\in K_{\text{st}}$, where $K_{\text{st}}=\{\lambda\in\R: \phi(\lambda)=0, \lambda\phi'(\lambda)>0\}$.