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

DOI10.1214/10-AOP601zbMATH Open1243.60066arXiv0907.5180MaRDI QIDQ653298

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Publication date: 9 January 2012

Published in: (Search for Journal in Brave)

Abstract: We consider a branching-selection system in $mathbbR$ 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 $Noinfty$, 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 $cgeqa$ 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.

Full work available at URL: https://arxiv.org/abs/0907.5180

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