Invariance principle for the maximal position process of branching Brownian motion in random environment

TL;DR

This paper investigates the maximal position process of branching Brownian motion in a random environment, establishing a strong law of large numbers and an invariance principle for the maximum particle position over time.

Contribution

It introduces a quenched strong law and an annealed invariance principle for the maximum position in branching Brownian motion within a random environment.

Findings

Max position satisfies a quenched strong law of large numbers.

Max position obeys an annealed invariance principle.

Results hold under certain conditions on the random environment.

Abstract

In this paper we study the maximal position process of branching Brownian motion in random spatial environment. The random environment is given by a process $\xi = \left(\xi(x)\right)_{x\in\mathbb{R}}$ satisfying certain conditions. We show that the maximum position $M_t$ of particles alive at time $t$ satisfies a quenched strong law of large numbers and an annealed invariance principle.