{
  "id": "1003.0348",
  "version": "v1",
  "published": "2010-03-01T13:43:40.000Z",
  "updated": "2010-03-01T13:43:40.000Z",
  "title": "On the Stochastic Heat Equation with Spatially-Colored Random forcing",
  "authors": [
    "Mohammud Foondun",
    "Davar Khoshnevisan"
  ],
  "comment": "+100 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the stochastic heat equation of the following form \\frac{\\partial}{\\partial t}u_t(x) = (\\sL u_t)(x) +b(u_t(x)) + \\sigma(u_t(x))\\dot{F}_t(x)\\quad \\text{for}t>0, x\\in \\R^d, where $\\sL$ is the generator of a L\\'evy process and $\\dot{F}$ is a spatially-colored, temporally white, gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data $u_0$ is a bounded and measurable function and $\\sigma$ is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also \\emph{weakly intermittent}. In addition, we study the same equation in the case that $\\mathcal{L}u$ is replaced by its massive/dispersive analogue $\\mathcal{L}u-\\lambda u$ where $\\lambda\\in\\R$. Furthermore, we extend our analysis to the case that the initial data $u_0$ is a measure rather than a function. As it turns out, the stochastic PDE in question does not have a mild solution in this case. We circumvent this problem by introducing a new concept of a solution that we call a \\emph{temperate solution}, and proceed to investigate the existence and uniqueness of a temperate solution. We are able to also give partial insight into the long-time behavior of the temperate solution when it exists and is unique. Finally, we look at the linearized version of our stochastic PDE, that is the case when $\\sigma$ is identically equal to one [any other constant works also].In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-03-01T13:43:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35R60"
    ],
    "keywords": [
      "stochastic heat equation",
      "spatially-colored random forcing",
      "mild solution",
      "initial data",
      "temperate solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 100,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.0348F"
    }
  }
}