Velocity of the L-branching Brownian motion
Résumé
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
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
L
-branching Brownian motion. We show that the position of the system grows linearly at a velocity vL
v
L
almost surely and we compute the asymptotic behavior of vL
v
L
as L
L
tends to infinity:
vL=2‾√−π222‾√L2+o(1L2),
v
L
2
π
2
2
2
L
2
o
1
L
2
as conjectured in [10]. The proof makes use of results by Berestycki, Berestycki and Schweinsberg [5] concerning branching Brownian motion in a strip.