# 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
Document type :
Preprints, Working Papers, ...
2010
Domain :
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-00514779
Contributor : Julien Berestycki <>
Submitted on : Friday, September 3, 2010 - 10:21:40 AM
Last modification on : Wednesday, October 12, 2016 - 1:02:28 AM

### Identifiers

• 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>

### Metrics

Consultations de la notice