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Preprints, Working Papers, ... Year : 2022

Branching Brownian motion conditioned on small maximum

Abstract

For a standard binary branching Brownian motion on the real line, it is known that the typical position of the maximal position $M_t$ among all particles alive at time $t$ is $m_t+\Theta(1)$ with $m_t=\sqrt{2}t-\frac{3}{2\sqrt{2}}\log t$. Further, it is proved independently in \cite{ABBS13} and \cite{ABK13} that the branching Brownian motion shifted by $m_t$ (or $M_t$) converges in law to some decorated Poisson point process. The goal of this work is to study the branching Brownian motion conditioned on $M_t\ll m_t$. We give a complete description of the limiting extremal process conditioned on $\{M_t\le \sqrt{2}\alpha t\}$ with $\alpha<1$, which reveals a phase transition at $\alpha=1-\sqrt{2}$. We also verify the conjecture of Derrida and Shi \cite{DS} on the precise asymptotic behaviour of $\P(M_t \leq \sqrt{2}\alpha t)$ for $\alpha < 1$.
On considère un mouvement Brownien branchant standard sur la ligne réelle. Il est connu que la plus grande position $M_t$ parmi les positions des particules en vie au temps $t$, translatée par $m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}}\log t$ converge en loi vers une variable de Gumbel aléatoirement translatée. Derrida et Shi (2017) ont conjecturé la forme du comportement asymptotique précis des estimées de grandes dévations inférieures $\mathbb{P}(M_t \leq \sqrt{2} \alpha t )$ pour $\alpha < 1$. On vérifie leur conjecture, et décrit la loi du mouvement Brownien branchant conditionné à avoir un petit maximum.
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Dates and versions

hal-02887692 , version 1 (18-11-2022)
hal-02887692 , version 2 (15-11-2023)

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Xinxin Chen, Hui He, Bastien Mallein. Branching Brownian motion conditioned on small maximum. 2022. ⟨hal-02887692v1⟩
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