# Survival of near-critical branching Brownian motion

Abstract : Consider a system of particles performing branching Brownian motion with negative drift $\mu = \sqrt{2 - \epsilon}$ and killed upon hitting zero. Initially there is one particle at $x>0$. Kesten showed that the process survives with positive probability if and only if $\epsilon>0$. Here we are interested in the asymptotics as $\eps\to 0$ of the survival probability $Q_\mu(x)$. It is proved that if $L= \pi/\sqrt{\epsilon}$ then for all $x \in \R$, $\lim_{\epsilon \to 0} Q_\mu(L+x) = \theta(x) \in (0,1)$ exists and is a travelling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when \$x
Type de document :
Pré-publication, Document de travail
2010
Domaine :
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-00514779
Contributeur : Julien Berestycki <>
Soumis le : vendredi 3 septembre 2010 - 10:21:40
Dernière modification le : lundi 29 mai 2017 - 14:24:27

### Identifiants

• HAL Id : hal-00514779, version 1
• ARXIV : 1009.0406

### Citation

Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg. Survival of near-critical branching Brownian motion. 2010. 〈hal-00514779〉

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