{
  "id": "1307.4496",
  "version": "v3",
  "published": "2013-07-17T03:56:47.000Z",
  "updated": "2015-05-19T09:30:23.000Z",
  "title": "Maximal displacement of a branching random walk in time-inhomogeneous environment",
  "authors": [
    "Bastien Mallein"
  ],
  "comment": "51 pages, to appear in SPA",
  "doi": "10.1016/j.spa.2015.05.011",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length $n$ of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time $n$. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order $n^{1/3}$, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other quantities, such as the consistent maximal displacement.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-17T12:02:53.000Z",
      "abstract": "In this article, we study a branching random walk evolving in a (macroscopically) time-inhomogeneous environment. We compute the first two orders of the asymptotic of the maximal displacement. The ballistic first order is given by the solution of an optimization problem, while the second term is of order $n^{1/3}$. This result partially answers a conjecture of Fang and Zeitouni. Additionally, we obtain an asymptotic for the so-called consistent maximal displacement of the process, which also is of order $n^{1/3}$.",
      "comment": "39 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-05-19T09:30:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60G50"
    ],
    "keywords": [
      "branching random walk",
      "time-inhomogeneous environment",
      "ballistic first order",
      "consistent maximal displacement",
      "second term"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.4496M"
    }
  }
}