{
  "id": "1307.0600",
  "version": "v3",
  "published": "2013-07-02T07:10:32.000Z",
  "updated": "2015-12-21T13:48:47.000Z",
  "title": "Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions",
  "authors": [
    "Le Chen",
    "Robert C. Dalang"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AOP954 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2015, Vol. 43, No. 6, 3006-3051",
  "doi": "10.1214/14-AOP954",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the nonlinear stochastic heat equation in the spatial domain $\\mathbb {R}$, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on $\\mathbb {R}$, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall's lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all $p$th moments $(p\\ge2)$ are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when $p=2$. We determine the growth indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152 (2012) 681-701].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-13T10:58:50.000Z",
      "abstract": "We study the nonlinear stochastic heat equation in the spatial domain $\\mathbb{R}$, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on $\\mathbb{R}$, such as the Dirac delta function, but this measure may also have non-compact support and even be non-tempered (for instance with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall's lemma, by keeping tight control over moments in the Picard iteration scheme. Upper and lower bounds on all $p$-th moments $(p\\ge 2)$ are obtained. These bounds become equalities for the parabolic Anderson model when $p=2$. We determine the growth indices introduced by Conus and Khoshnevisan.",
      "comment": "Revised version to appear in Ann. Probab. The organization of the paper has been changed. 45 pages, 1 figure. arXiv admin note: text overlap with arXiv:1210.1690",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-12-21T13:48:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "10H15",
      "60H60",
      "35H60"
    ],
    "keywords": [
      "nonlinear stochastic heat equation",
      "rough initial conditions",
      "growth indices",
      "parabolic anderson model",
      "central special case"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.0600C"
    }
  }
}