The extremal point process of branching brownian motion in ℝ^d

We consider a branching Brownian motion in R^d with d ≥ 1 in which the position X^(u) _t ∈ R^d of a particle u at time t can be encoded by its direction θ^(u) _t ∈ S^d−1 and its distance R^(u) _t to 0. We prove that the extremal point process Σδ_{(θ^(u) t,R^(u) t −m^(d) t )} (where the sum is over all particles alive at time t and m^(d)_t is an explicit centering term) converges in distribution to a randomly shifted, decorated Poisson point process on Sau4:^d−1 ×R. More precisely, the so-called clan-leaders form a Cox process with intensity proportional to D_∞(θ)e^−√2r dr dθ, 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 Stasinski, 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).