{
  "id": "math/0507585",
  "version": "v3",
  "published": "2005-07-28T12:37:59.000Z",
  "updated": "2007-07-25T11:49:57.000Z",
  "title": "Geometric characterization of intermittency in the parabolic Anderson model",
  "authors": [
    "Jürgen Gärtner",
    "Wolfgang König",
    "Stanislav Molchanov"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000764 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2007, Vol. 35, No. 2, 439-499",
  "doi": "10.1214/009117906000000764",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the parabolic Anderson problem $\\partial_tu=\\Delta u+\\xi(x)u$ on $\\mathbb{R}_+\\times\\mathbb{Z}^d$ with localized initial condition $u(0,x)=\\delta_0(x)$ and random i.i.d. potential $\\xi$. Under the assumption that the distribution of $\\xi(0)$ has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as $t\\to\\infty$, the overwhelming contribution to the total mass $\\sum_xu(t,x)$ comes from a slowly increasing number of ``islands'' which are located far from each other. These ``islands'' are local regions of those high exceedances of the field $\\xi$ in a box of side length $2t\\log^2t$ for which the (local) principal Dirichlet eigenvalue of the random operator $\\Delta+\\xi$ is close to the top of the spectrum in the box. We also prove that the shape of $\\xi$ in these regions is nonrandom and that $u(t,\\cdot)$ is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-07-25T11:49:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H25",
      "82C44",
      "60F10",
      "35B40"
    ],
    "keywords": [
      "parabolic anderson model",
      "geometric characterization",
      "intermittency",
      "parabolic anderson problem",
      "principal dirichlet eigenvalue"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7585G"
    }
  }
}