Open Access
Translator Disclaimer
November 2011 Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations
Rick Durrett, Daniel Remenik
Ann. Probab. 39(6): 2043-2078 (November 2011). DOI: 10.1214/10-AOP601

Abstract

We consider a branching-selection system in 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 → ∞, 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 ca 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.

Citation

Download Citation

Rick Durrett. Daniel Remenik. "Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations." Ann. Probab. 39 (6) 2043 - 2078, November 2011. https://doi.org/10.1214/10-AOP601

Information

Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1243.60066
MathSciNet: MR2932664
Digital Object Identifier: 10.1214/10-AOP601

Subjects:
Primary: 35C07 , 35R35 , 60F99 , 60J80 , 60J99

Keywords: Branching random walk , Branching-selection system , free boundary equation , traveling wave solutions , Wiener–Hopf equation

Rights: Copyright © 2011 Institute of Mathematical Statistics

JOURNAL ARTICLE
36 PAGES


SHARE
Vol.39 • No. 6 • November 2011
Back to Top