The Annals of Probability
- Ann. Probab.
- Volume 39, Number 6 (2011), 2043-2078.
Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations
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 c ≥ 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.
Ann. Probab., Volume 39, Number 6 (2011), 2043-2078.
First available in Project Euclid: 17 November 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J99: None of the above, but in this section 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 35R35: Free boundary problems 35C07: Traveling wave solutions 60F99: None of the above, but in this section
Durrett, Rick; Remenik, Daniel. Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations. Ann. Probab. 39 (2011), no. 6, 2043--2078. doi:10.1214/10-AOP601. https://projecteuclid.org/euclid.aop/1321539116