The extremal point process of branching Brownian motion in Rd

Abstract

We consider a branching Brownian motion in ${\mathbb{R}}^{\mathit{d}}$ with $\mathit{d}\ge 1$ in which the position ${\mathit{X}}_{\mathit{t}}^{(\mathit{u})}\in {\mathbb{R}}^{\mathit{d}}$ of a particle u at time t can be encoded by its direction ${\mathit{\theta }}_{\mathit{t}}^{(\mathit{u})}\in {\mathbb{S}}^{\mathit{d}-1}$ and its distance ${\mathit{R}}_{\mathit{t}}^{(\mathit{u})}$ to 0. We prove that the extremal point process $\sum {\mathit{\delta }}_{({\mathit{\theta }}_{\mathit{t}}^{(\mathit{u})},{\mathit{R}}_{\mathit{t}}^{(\mathit{u})}-{\mathit{m}}_{\mathit{t}}^{(\mathit{d})})}$ (where the sum is over all particles alive at time t and ${\mathit{m}}_{\mathit{t}}^{(\mathit{d})}$ is an explicit centering term) converges in distribution to a randomly shifted, decorated Poisson point process on ${\mathbb{S}}^{\mathit{d}-1}\times \mathbb{R}$. More precisely, the so-called clan-leaders form a Cox process with intensity proportional to ${\mathit{D}}_{\infty }(\mathit{\theta }){\mathit{e}}^{-\sqrt{2}\mathit{r}}\mathrm{d}\mathit{r}\mathrm{d}\mathit{\theta }$, where ${\mathit{D}}_{\infty }(\mathit{\theta })$ is the limit of the derivative martingale in direction θ 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. Henri Poincaré Probab. Stat. 57 (2021) 1786–1810). The proof builds on that paper and on Kim, Lubetzky and Zeitouni (Ann. Appl. Probab. 33 (2023) 1315–1368).