Branching Random Walk in an inhomogeneous breeding potential

TL;DR

This paper studies a branching random walk with inhomogeneous breeding potential, showing explosion occurs for p>1 and analyzing the asymptotic behavior of the rightmost particle when it doesn't explode.

Contribution

It establishes conditions for explosion in the process and characterizes the asymptotic behavior of the extremal particles in non-explosive cases.

Findings

Population explodes almost surely if p > 1.

Population does not explode if p ≤ 1.

Asymptotic behavior of the rightmost particle is determined in non-explosive cases.

Abstract

We consider a continuous-time branching random walk in the inhomogeneous breeding potential $\beta|.|^p$, where $\beta > 0$, $p \geq 0$. We prove that the population almost surely explodes in finite time if $p > 1$ and doesn't explode if $p \leq 1$. In the non-explosive cases, we determine the asymptotic behaviour of the rightmost particle.