{
  "id": "2206.02641",
  "version": "v1",
  "published": "2022-06-06T14:23:49.000Z",
  "updated": "2022-06-06T14:23:49.000Z",
  "title": "Necessary and sufficient conditions to solve parabolic Anderson model with rough noise",
  "authors": [
    "Shuhui Liu",
    "Yaozhong Hu",
    "Xiong Wang"
  ],
  "comment": "45 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We obtain necessary and sufficient condition for the existence of $n$-th chaos of the solution to the parabolic Anderson model $\\frac{\\partial}{\\partial t}u(t,x)=\\frac{1}{2}\\Delta u(t,x)+u(t,x)\\dot{W}(t,x)$, where $\\dot{W}(t,x)$ is a fractional Brownian field with temporal Hurst parameter $H_0\\ge 1/2$ and spatial parameters $H$ $ =(H_1, \\cdots, H_d)$ $ \\in (0, 1)^d$. When $d=1$, we extend the condition on the parameters under which the chaos expansion of the solution is convergent in the mean square sense, which is both sufficient and necessary under some circumstances.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-06T14:23:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H05",
      "60H07",
      "26D15"
    ],
    "keywords": [
      "parabolic anderson model",
      "sufficient condition",
      "rough noise",
      "fractional brownian field",
      "temporal hurst parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}