The maximum of branching Brownian motion in $\mathbb{R}^d$

TL;DR

This paper proves that in multi-dimensional branching Brownian motion, the distribution of the maximum particle distance from the origin converges to a shifted Gumbel distribution as time progresses.

Contribution

It establishes the limiting distribution of the maximum particle distance in multi-dimensional BBM, extending known results from one dimension to higher dimensions.

Findings

The maximum distance law converges to a shifted Gumbel distribution.

The convergence holds for dimensions d ≥ 2.

Provides a probabilistic description of extremal particles in multi-dimensional BBM.

Abstract

We show that in branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$, the law of $R_t^*$, the maximum distance of a particle from the origin at time $t$, converges as $t\to\infty$ to the law of a randomly shifted Gumbel random variable.