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 [10] 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: \[v_L = \sqrt{2} - \frac{\pi ^2} {2\sqrt{2} L^2} + o\left({\frac {1}{L^2}}\right) ,\] as conjectured in [10]. The proof makes use of results by Berestycki, Berestycki and Schweinsberg [5] concerning branching Brownian motion in a strip.