## Abstract

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).

Original language | English (US) |
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Pages (from-to) | 955-982 |

Number of pages | 28 |

Journal | Annals of Probability |

Volume | 52 |

Issue number | 3 |

DOIs | |

State | Published - 2024 |

## Keywords

- Branching Brownian motion
- cluster process
- decorated Poisson point process
- extremal point process
- extreme value theory

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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