{
  "id": "2002.07142",
  "version": "v1",
  "published": "2020-02-17T18:57:34.000Z",
  "updated": "2020-02-17T18:57:34.000Z",
  "title": "The continuum parabolic Anderson model with a half-Laplacian and periodic noise",
  "authors": [
    "Alexander Dunlap"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $\\partial_t u=-(-\\Delta)^{1/2}u+\\xi u$, where $\\xi$ is a periodic spatial white noise. To be precise, we construct limits as $\\varepsilon\\to 0$ to solutions of $\\partial_t u_{\\varepsilon}=-(-\\Delta)^{1/2}u_{\\varepsilon}+(\\xi_{\\varepsilon}-C_{\\varepsilon})u_{\\varepsilon}$, where $\\xi_{\\varepsilon}$ is a mollification of $\\xi$ at scale $\\varepsilon$ and $C_{\\varepsilon}$ is a logarithmically diverging renormalization constant. We use a simple renormalization scheme based on that of Hairer and Labb\\'e, \"A simple construction of the continuum parabolic Anderson model on $\\mathbf{R}^{2}$.\"",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-17T18:57:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15"
    ],
    "keywords": [
      "continuum parabolic anderson model",
      "periodic noise",
      "continuum fractional parabolic anderson model",
      "renormalized continuum fractional parabolic anderson",
      "periodic spatial white noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}