{
  "id": "2109.03875",
  "version": "v1",
  "published": "2021-09-08T18:51:13.000Z",
  "updated": "2021-09-08T18:51:13.000Z",
  "title": "Quantitative central limit theorems for the parabolic Anderson model driven by colored noises",
  "authors": [
    "David Nualart",
    "Panqiu Xia",
    "Guangqu Zheng"
  ],
  "comment": "51 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use It\\^o's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincar\\'e inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (2020 \\emph{Electron. J. Probab.}) and Nualart-Song-Zheng (2021 \\emph{ALEA, Lat. Am. J. Probab. Math. Stat.}).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-08T18:51:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H07",
      "60H15",
      "60F05"
    ],
    "keywords": [
      "parabolic anderson model driven",
      "colored noises",
      "martingale structure",
      "second-order gaussian poincare inequality",
      "establish quantitative central limit theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}