{
  "id": "1102.4921",
  "version": "v1",
  "published": "2011-02-24T08:07:19.000Z",
  "updated": "2011-02-24T08:07:19.000Z",
  "title": "A two cities theorem for the parabolic Anderson model",
  "authors": [
    "Wolfgang König",
    "Hubert Lacoin",
    "Peter Mörters",
    "Nadia Sidorova"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AOP405 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2009, Vol. 37, No. 1, 347-392",
  "doi": "10.1214/08-AOP405",
  "categories": [
    "math.PR"
  ],
  "abstract": "The parabolic Anderson problem is the Cauchy problem for the heat equation $\\partial_tu(t,z)=\\Delta u(t,z)+\\xi(z)u(t,z)$ on $(0,\\infty)\\times {\\mathbb{Z}}^d$ with random potential $(\\xi(z):z\\in{\\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $\\xi(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-24T08:07:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic anderson model",
      "cities theorem",
      "parabolic anderson problem",
      "weak limit theorem",
      "cauchy problem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.4921K"
    }
  }
}