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

We consider a branching Brownian motion in Rd with d ≥ 1 in which the position X t ∈ Rd of a particle u at time t can be encoded by its direction θ t ∈ Sd−1 and its distance R (u) t to 0. We prove that the extremal point process ∑ δ θ (u) t ,R (u) t −m(d) t (where the sum is over all particles alive at time t and m t is an explicit centring 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. H. Poincaré 57:1786–1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).