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 X(u) t is an element of Rd of a particle u at time t can be encoded by its direction theta(u) t is an element of Sd-1 and its distance R(u) t to 0. We prove that the extremal point process Sigma delta(theta(u) t,Rt(u)-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 Sd-1 x R. More precisely, the so-called clan -leaders form a Cox process with intensity proportional to root D infinity(theta)e- 2r dr d theta, where D infinity (theta) is the limit of the derivative martingale in direction theta 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 Stasin<acute accent>ski, Berestycki and Mallein (Ann. Inst. Henri Poincar & eacute; 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).