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.
Highlights
The branching Brownian motion is a branching Markov process whose study dates back to [23]
Each particle moves according to a Brownian motion, during an exponentially distributed time and splits into two new particles, which start the same process from their place of birth
For each ξ ∈ C, ξ = (ξ1, . . . , ξn), we define the branching Brownian motion starting from this configuration as before, but with n particles at time 0 positioned at ξ1, . . . , ξn
Summary
The branching Brownian motion (or BBM) is a branching Markov process whose study dates back to [23] It has been the subject of a large literature, especially for its connection with the F-KPP equation, highlighted by McKean [29]. The selection tends to eliminate the lowest particles, that have a too small fitness value by comparison with the best ones. We consider a system of particles evolving as before, but where in addition a particle dies as soon as it is at a distance L of the highest particle alive at the same time. This system is called the L-branching Brownian motion or L-BBM
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