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

TL;DR

This paper studies a branching-selection particle system, showing its large-scale behavior converges to a deterministic process described by a free boundary integro-differential equation, with solutions linked to Wiener-Hopf equations and traveling wave phenomena.

Contribution

It establishes the convergence of the particle system to a deterministic measure-valued process and characterizes the existence of traveling wave solutions via Wiener-Hopf equations.

Findings

Convergence of empirical measure to a deterministic process.

Existence and uniqueness of traveling wave solutions depending on speed.

Connection between solutions 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 $c\geq a$ 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.