The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential

TL;DR

This paper investigates the almost-sure growth rate of a branching Brownian motion with a quadratic breeding potential, revealing asymptotic behavior despite finite expected population explosion.

Contribution

It provides the first asymptotic characterization of the almost-sure population growth rate in BBM with quadratic breeding rate.

Findings

Population size grows exponentially almost surely

Expected population size explodes in finite time

Asymptotic growth rate determined

Abstract

In this note we consider a branching Brownian motion (BBM) on $\mathbb{R}$ in which a particle at spatial position $y$ splits into two at rate $\beta y^2$, where $\beta>0$ is a constant. This is a critical breeding rate for BBM in the sense that the expected population size blows up in finite time while the population size remains finite, almost surely, for all time. We find an asymptotic for the almost sure rate of growth of the population.