The extremal point process of branching brownian motion in ℝd

Julien Berestycki, Yujin H. Kim, Eyal Lubetzky, Bastien Mallein, Ofer Zeitouni

Research output: Contribution to journalArticlepeer-review


We consider a branching Brownian motion in Rd with d ≥ 1 in which the position X(u) t ∈ Rd of a particle u at time t can be encoded by its direction θ(u) t ∈ Sd−1 and its distance R(u) t to 0. We prove that the extremal point process Σδ(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 languageEnglish (US)
Pages (from-to)955-982
Number of pages28
JournalAnnals of Probability
Issue number3
StatePublished - 2024


  • 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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