{
  "id": "0805.0557",
  "version": "v1",
  "published": "2008-05-05T15:42:27.000Z",
  "updated": "2008-05-05T15:42:27.000Z",
  "title": "Intermittence and nonlinear parabolic stochastic partial differential equations",
  "authors": [
    "Mohammud Foondun",
    "Davar Khoshnevisan"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider nonlinear parabolic SPDEs of the form $\\partial_t u=\\sL u + \\sigma(u)\\dot w$, where $\\dot w$ denotes space-time white noise, $\\sigma:\\R\\to\\R$ is [globally] Lipschitz continuous, and $\\sL$ is the $L^2$-generator of a L\\'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\\sL$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\\sL$ in dimension $(1+1)$. When $\\sL=\\kappa\\partial_{xx}$ for $\\kappa>0$, these formulas agree with the earlier results of statistical physics \\cite{Kardar,KrugSpohn,LL63}, and also probability theory \\cite{BC,CM94} in the two exactly-solvable cases where $u_0=\\delta_0$ and $u_0\\equiv 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-05-05T15:42:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "82B44"
    ],
    "keywords": [
      "nonlinear parabolic stochastic partial differential",
      "parabolic stochastic partial differential equations",
      "denotes space-time white noise",
      "upper second-moment liapounov exponent",
      "intermittence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0805.0557F"
    }
  }
}