Hydrodynamics of the $N$-BBM process

Abstract

The Branching Brownian Motions (BBM) are particles performing independent Brownian motions in $\mathbb{R}$ and each particle at rate 1 creates a new particle at her current position; the newborn particle increments and branchings are independent of the other particles. The $N$-BBM starts with $N$ particles and at each branching time, the leftmost particle is removed so that the total number of particles is $N$ for all times. The $N$-BBM was proposed by Maillard and belongs to a family of processes introduced by Brunet and Derrida. We fix a density $\rho$ with a left boundary $L=\sup \{r\in \mathbb{R}:{\int }_r^{\infty }\rho (x)dx=1\}>-\infty$ and let the initial particle positions be iid continuous random variables with density $\rho$. We show that the empirical measure associated to the particle positions at a fixed time $t$ converges to an absolutely continuous measure with density $\psi (\cdot ,t)$, as $N\to \infty$. The limit $\psi$ is solution of a free boundary problem (FBP) when this solution exists. The existence of solutions for finite time-intervals has been recently proved by Lee.

25 pages, 2 figures