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

Abstract : We consider a branching Brownian motion in $\R^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \R^d$ of a particle $u$ at time $t$ can be encoded by its direction $\theta^{(u)}_t \in \S^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the \emph{extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m^{(d)}_t$ is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on $\S^{d-1} \times \R$. More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to $D_\infty(\theta) e^{-\sqrt{2}r} \d r\d \theta $, where $D_\infty(\theta)$ 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 Stasi\'nski, Berestycki and Mallein (Ann.\ Inst.\ H.\ Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).
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https://hal.archives-ouvertes.fr/hal-03514801
Contributor : Bastien Mallein Connect in order to contact the contributor
Submitted on : Thursday, January 6, 2022 - 2:18:39 PM
Last modification on : Thursday, January 13, 2022 - 4:12:24 AM
Long-term archiving on: : Thursday, April 7, 2022 - 7:24:00 PM

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  • HAL Id : hal-03514801, version 1

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Julien Berestycki, Yujin Kim, Eyal Lubetzky, Bastien Mallein, Ofer Zeitouni. The extremal point process of branching Brownian motion in $\R^d$. 2022. ⟨hal-03514801⟩

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