The extremal point process of branching Brownian motion in Rd

We consider a branching Brownian motion in Rd with d≥1 in which the position Xt(u)∈Rd of a particle u at time t can be encoded by its direction θt(u)∈Sd−1 and its distance Rt(u) to 0. We prove that the extremal point process ∑δ (θt(u),Rt(u)−mt(d)) (where the sum is over all particles alive at time t and mt(d) is an explicit centering term) converges in distribution to a randomly shifted, decorated Poisson point process on Sd−1×R. More precisely, the so-called clan-leaders form a Cox process with intensity proportional to D∞(θ)e− 2rdrdθ, where D∞(θ) 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).