{
  "id": "0907.5180",
  "version": "v5",
  "published": "2009-07-29T17:18:03.000Z",
  "updated": "2023-04-17T23:13:37.000Z",
  "title": "Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations",
  "authors": [
    "Rick Durrett",
    "Daniel Remenik"
  ],
  "comment": "A mistake in one of the proofs was pointed out to us, but the gap in the argument has been addressed in a recent paper, see Remarks 1.1 and 2.10",
  "journal": "Annals of Probability 2011, Vol. 39, No. 6, 2043-2078",
  "doi": "10.1214/10-AOP601",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a branching-selection system in $\\mathbb {R}$ 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 $N\\to\\infty$, 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 $c\\geq a$ 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.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-01-12T06:44:00.000Z",
      "comment": "Published in at http://dx.doi.org/10.1214/10-AOP601 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)"
    },
    {
      "version": "v5",
      "updated": "2023-04-17T23:13:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free boundary problems",
      "brunet-derrida particle systems",
      "wiener-hopf equations",
      "traveling wave solution traveling",
      "leftmost particle"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.5180D"
    }
  }
}