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

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• Daniel Remenik, Cornell University
• Tuesday 12 January 2010, 16:30-17:30
• MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.

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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 tends to infinity, 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. This is joint work with Rick Durrett.

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