Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations

Abstract

We consider a branching-selection system in $\mathbb{R}$ with $N$ particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as $N\to \infty$, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed $c$ or no such solution depending on whether $c\ge a$ or $c<a$, where $a$ is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations.

A mistake in one of the proofs was pointed out to us, but the gap in the argument has been addressed in a recent paper, see Remarks 1.1 and 2.10

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• Evolution in predator-prey systems
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