# Velocity of the $L$-branching Brownian motion

Abstract : We consider a branching-selection system of particles on the real line that evolves according to the following rules: each particle moves according to a Brownian motion during an exponential lifetime and then splits into two new particles and, when a particle is at a distance $L$ of the highest particle, it dies without splitting. This model has been introduced by Brunet, Derrida, Mueller and Munier in the physics literature and is called the $L$-branching Brownian motion. We show that the position of the system grows linearly at a velocity $v_L$ almost surely and we compute the asymptotic behavior of $v_L$ as $L$ tends to infinity: $v_L = \sqrt{2} − \pi^2 / 2 \sqrt{2} L^2 + o(1/L^2)$, as conjectured by Brunet, Derrida, Mueller and Munier. The proof makes use of results by Berestycki, Berestycki and Schweinsberg concerning branching Brownian motion in a strip.
Keywords :
Type de document :
Pré-publication, Document de travail
32 pages, 6 figures, to appear in the Electronic Journal of Probability. 2015
Domaine :

https://hal.archives-ouvertes.fr/hal-01214605
Contributeur : Michel Pain <>
Soumis le : jeudi 7 avril 2016 - 10:39:52
Dernière modification le : lundi 29 mai 2017 - 14:25:11

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Michel_Pain_L-BBM_v2.pdf
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### Identifiants

• HAL Id : hal-01214605, version 2
• ARXIV : 1510.02683

### Citation

Michel Pain. Velocity of the $L$-branching Brownian motion. 32 pages, 6 figures, to appear in the Electronic Journal of Probability. 2015. <hal-01214605v2>

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