{
  "id": "2202.08636",
  "version": "v1",
  "published": "2022-02-17T12:53:41.000Z",
  "updated": "2022-02-17T12:53:41.000Z",
  "title": "Parabolic Anderson model on critical Galton-Watson trees in a Pareto environment",
  "authors": [
    "Eleanor Archer",
    "Anne Pein"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \\to \\infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier results on $\\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-17T12:53:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "35K40",
      "60J27"
    ],
    "keywords": [
      "parabolic anderson model",
      "critical galton-watson tree",
      "pareto environment",
      "extra spatial randomness",
      "pareto potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}