Branching Brownian Motion with catalytic branching at the origin

TL;DR

This paper studies a special branching Brownian motion where particles only split at the origin, analyzing its long-term behavior, the speed of the rightmost particle, and establishing a Strong Law of Large Numbers for this process.

Contribution

It introduces and analyzes a catalytic branching Brownian motion model with branching only at the origin, providing new asymptotic results and laws of large numbers.

Findings

Asymptotic behavior of particle counts above λt for λ > 0

Almost sure asymptotic speed of the rightmost particle

Strong Law of Large Numbers for the process

Abstract

We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate \beta > 0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above \lambda t at time t, for \lambda > 0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.