{
  "id": "2101.05997",
  "version": "v1",
  "published": "2021-01-15T07:26:53.000Z",
  "updated": "2021-01-15T07:26:53.000Z",
  "title": "Solvability of parabolic Anderson equation with fractional Gaussian noise",
  "authors": [
    "Zhen-Qing Chen",
    "Yaozhong Hu"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model $\\frac{\\partial u}{\\partial t}=\\frac{1}{2}\\frac{\\partial^2 u}{\\partial x^2}+u\\dot{W}$ on $[0, \\infty)\\times {\\bf R}^d $ with $d\\geq 1$ has a unique random field solution, where $W(t, x)$ is a fractional Brownian sheet on $[0, \\infty)\\times {\\bf R}^d$ and formally $\\dot W =\\frac{\\partial^{d+1}}{\\partial t \\partial x_1 \\cdots \\partial x_d} W(t, x)$. When the noise $W(t, x)$ is white in time, our condition is both necessary and sufficient when the initial data $u(0, x)$ is bounded between two positive constants. When the noise is fractional in time with Hurst parameter $H_0>1/2$, our sufficient condition, which improves the known results in literature, is different from the necessary one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-15T07:26:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60G60",
      "60G15",
      "60G22",
      "35R60"
    ],
    "keywords": [
      "fractional gaussian noise",
      "parabolic anderson equation",
      "sufficient condition",
      "solvability",
      "continuous time parabolic anderson model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}