Velocity of the $L$-branching Brownian motion - Laboratoire de Probabilités et Modèles Aléatoires
Journal Articles Electronic Journal of Probability Year : 2016

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.
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Dates and versions

hal-01214605 , version 1 (12-10-2015)
hal-01214605 , version 2 (07-04-2016)
hal-01214605 , version 3 (13-11-2023)

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Cite

Michel Pain. Velocity of the $L$-branching Brownian motion. Electronic Journal of Probability, 2016, 24, Paper No. 28, 28. ⟨10.1214/16-EJP4639⟩. ⟨hal-01214605v3⟩
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