Maximum of Branching Brownian Motion among mild obstacles

TL;DR

This paper investigates the maximum particle height in a one-dimensional branching Brownian motion with obstacles, revealing asymptotic behavior and the influence of obstacle configuration on particle paths.

Contribution

It provides the first almost sure asymptotics for the maximum height in branching Brownian motion with space-dependent branching rates and obstacles.

Findings

Asymptotic behavior of the maximum particle height is characterized.

The path of the extremal particle is described in relation to obstacle placement.

Dependence of maximum height on obstacle size and location is established.

Abstract

We study the height of the maximal particle at time $t$ of a one dimensional branching Brownian motion with a space-dependent branching rate. The branching rate is set to zero in finitely many intervals (obstacles) of order $t$. We obtain almost sure asymptotics of the first order of the maximum, describe the path of a particle reaching this height and describe its dependence on the size and location of the obstacles.