| Summary: | 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 Pδ (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 S d−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 Stasinski, Berestycki and Mallein (Ann. Inst. H. ´ Poincaré 57:1786–1810, 2021). The proof builds on that paper and on Kim, Lubetzky and Zeitouni (Ann. Appl. Prob. 33(2):1315–1368, 2023).
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