{
  "id": "2205.13105",
  "version": "v1",
  "published": "2022-05-26T01:55:15.000Z",
  "updated": "2022-05-26T01:55:15.000Z",
  "title": "Central limit theorems for heat equation with time-independent noise: the regular and rough cases",
  "authors": [
    "Raluca M. Balan",
    "Wangjun Yuan"
  ],
  "comment": "43 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension $d\\geq 1$, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index $H \\in (\\frac{1}{4},\\frac{1}{2})$ in dimension $d=1$. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-26T01:55:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "time-independent noise",
      "rough cases",
      "heat equation",
      "spatial integral",
      "quantitative central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}