{
  "id": "2403.01565",
  "version": "v2",
  "published": "2024-03-03T17:02:44.000Z",
  "updated": "2024-03-11T12:05:00.000Z",
  "title": "Strong survival and extinction for branching random walks via a new order for generating functions",
  "authors": [
    "Daniela Bertacchi",
    "Fabio Zucca"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider general discrete-time branching random walks on a countable set $X$. According to these processes, a particle at $x\\in X$ generates a random number of children and places them at (some of) the sites of $X$, not necessarily independently nor with the same law at different starting vertices $x$.We introduce a new type of stochastic ordering of branching random walks, generalizing the germ order introduced by Hutchcroft in arXiv:2011.06402, which relies on the generating function of the process. We prove that given two branching random walks with law $\\mathbf{\\mu}$ and $\\mathbf{\\nu}$ respectively, with $\\mathbf{\\mu} \\ge \\mathbf{\\nu}$, then in every set where there is extinction according to $\\mathbf{\\mu}$, there is extinction also according to $\\mathbf{\\nu}$. Moreover, in every set where there is strong local survival according to $\\mathbf{\\nu}$, there is strong local survival also according to $\\mathbf{\\nu}$, provided that the supremum of the global extinction probabilities, for the $\\mathbf{\\nu}$-process, taken over all starting points $x$, is strictly smaller than 1. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a branching random walk is the only fixed point of its generating function, whose supremum over all starting coordinates, may be equal to 1.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-03-11T12:05:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80"
    ],
    "keywords": [
      "generating function",
      "strong survival",
      "strong local survival",
      "global extinction probability",
      "general discrete-time branching random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}