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,wherea 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. © 2011 Institute of Mathematical Statistics.

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Author Richard Timothy Durrett Mathematics