{
  "id": "1507.05385",
  "version": "v1",
  "published": "2015-07-20T05:17:49.000Z",
  "updated": "2015-07-20T05:17:49.000Z",
  "title": "On weak convergence of stochastic heat equation with colored noise",
  "authors": [
    "Pavel Bezdek"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this work we are going to show weak convergence of a probability measure corresponding to the solution of the following nonlinear stochastic heat equation $\\frac{\\partial}{\\partial t} u_{t}(x) = \\frac{\\kappa}{2} \\Delta u_{ t}(x) + \\sigma(u_{t}(x))\\eta_\\alpha$ with colored noise $\\eta_\\alpha$ to the measure corresponding to the solution of the same equation but with white noise $\\eta$ as $\\alpha \\uparrow 1$ on the space of continuous functions with compact support. The noise $\\eta_\\alpha$ is assumed to be colored in space and its covariance is given by $\\operatorname{E} \\left[ \\eta_\\alpha(t,x) \\eta_\\alpha(s,y) \\right] = \\delta(t-s) f_\\alpha(x-y)$ where $f_\\alpha$ is the Riesz kernel $f_\\alpha(x) \\propto 1/\\left|x\\right|^\\alpha$. We will also state a result about continuity of measure in $\\alpha$, for $\\alpha \\in (0,1)$. We will work with the classical notion of weak convergence of measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-20T05:17:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak convergence",
      "colored noise",
      "nonlinear stochastic heat equation",
      "compact support",
      "measure corresponding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}