The extremal point process of branching Brownian motion in $\mathbb{R}^d$

TL;DR

This paper characterizes the limiting distribution of the extremal particles in a multi-dimensional branching Brownian motion, showing it converges to a decorated Poisson point process with a specific intensity related to the derivative martingale.

Contribution

It proves the conjecture that the extremal process in $R^d$ converges to a decorated Poisson process, extending known 1D results to higher dimensions.

Findings

The extremal point process converges to a decorated Poisson process.

The clan-leaders form a Cox process with a specific intensity.

The limiting decoration process is related to the 1D case.

Abstract

We consider a branching Brownian motion in $\mathbb{R}^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb{R}^d$ of a particle $u$ at time $t$ can be encoded by its direction $\theta^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m^{(d)}_t$ is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on $\mathbb{S}^{d-1} \times \mathbb{R}$. More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to $D_\infty(\theta) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} \theta $, where $D_\infty(\theta)$ is the limit of the derivative martingale in direction $\theta$ and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasi\'nski, Berestycki and Mallein (Ann. Inst. H. Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).