{
  "id": "0911.2227",
  "version": "v1",
  "published": "2009-11-11T21:00:44.000Z",
  "updated": "2009-11-11T21:00:44.000Z",
  "title": "The critical random barrier for the survival of branching random walk with absorption",
  "authors": [
    "Bruno Jaffuel"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a branching random walk on $\\r$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. \\cite{BLSW91} determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a particle in a given generation, we add a second order term $a n^{1/3}$ to the position of the barrier for the $n^\\mathrm{th}$ generation and find an explicit critical value $a_c$ such that the process dies when $aa_c$. We also obtain the rate of extinction when $a < a_c$ and a lower bound on the surviving population when $a > a_c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-11T21:00:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80"
    ],
    "keywords": [
      "branching random walk",
      "critical random barrier",
      "absorption",
      "generation",
      "second order term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.2227J"
    }
  }
}