Branching Brownian motion with "mild" Poissonian obstacles

TL;DR

This paper analyzes a spatial branching process influenced by random 'mild' obstacles, revealing how local and global growth rates behave under various conditions and providing asymptotic results with correction terms.

Contribution

It introduces a detailed analysis of branching Brownian motion with mild Poissonian obstacles, including growth rate characterizations and asymptotic behaviors across different dimensions.

Findings

Quenched local growth rate equals the free branching rate.

Dichotomy in local growth for arbitrary diffusions, independent of obstacle intensity.

Asymptotic global growth rates with subexponential correction terms.

Abstract

We study a spatial branching model, where the underlying motion is Brownian motion and the branching is affected by a random collection of reproduction blocking sets called "mild" obstacles. We show that the quenched local growth rate is given by the branching rate in the `free' region . When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the local growth that is independent of the Poissonian intensity. Finally, and most importantly, we obtain the asymptotics (in probability) of the quenched (when $d\le 2$) and the annealed (arbitrary d) global growth rates, and identify subexponential correction terms.