Branching Brownian motion in an expanding ball and application to the mild obstacle problem

TL;DR

This paper investigates the behavior of branching Brownian motion in environments with mild Poissonian obstacles and within expanding balls, providing large deviation bounds and strong laws of large numbers for the process.

Contribution

It introduces quenched large deviation bounds and a strong law for BBM among mild obstacles, and analyzes BBM in expanding balls with reactivation, advancing understanding of these stochastic processes.

Findings

Established upper bounds on large-deviation probabilities for BBM mass among obstacles.

Proved a strong law of large numbers for the total mass in the obstacle environment.

Derived large-deviation results for BBM inside expanding, reactivatable balls.

Abstract

We first study a $d$-dimensional branching Brownian motion (BBM) among mild Poissonian obstacles, where a random trap field in $\mathbb{R}^d$ is created via a Poisson point process. The trap field consists of balls of fixed radius centered at the atoms of the Poisson point process. The mild obstacle rule is that when particles are inside traps, they branch at a lower rate, which is allowed to be zero, whereas when outside traps they branch at the normal rate. We prove upper bounds on the large-deviation probabilities for the total mass of BBM among mild obstacles, which we then use along with the Borel-Cantelli lemma to prove the corresponding strong law of large numbers. Our results are quenched, that is, they hold in almost every environment with respect to the Poisson point process. Our strong law improves on the existing corresponding weak law in [6]. Then, we study a $d$-dimensional BBM inside subdiffusively expanding balls, where the boundary of the ball is deactivating in the sense that once a particle of the BBM hits the moving boundary, it is instantly deactivated but can be reactivated at a later time provided its ancestral line is fully inside the expanding ball at that later time. We obtain a large-deviation result as time tends to infinity on the probability that the mass inside the ball is aytpically small. An essential ingredient in the proofs of the mild obstacle problem turns out to be the large-deviation result on the mass of BBM inside expanding balls.