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

Abstract

We consider a branching Brownian motion in ${\mathbb{R}}^d$ with $d\ge 1$ in which the position $X_t^{(u)}\in {\mathbb{R}}^d$ of a particle $u$ at time $t$ can be encoded by its direction ${\theta }_t^{(u)}\in {\mathbb{S}}^{d-1}$ and its distance $R_t^{(u)}$ to 0. We prove that the {\it extremal point process} $\sum {\delta }_{{\theta }_t^{(u)},R_t^{(u)}-m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m_t^{(d)}$ 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 Stasiński, Berestycki and Mallein (Ann. Inst. H. Poincaré 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).

20 pages, 4 figures; Version 2 includes referees comments