{
  "id": "2207.13781",
  "version": "v1",
  "published": "2022-07-27T20:39:27.000Z",
  "updated": "2022-07-27T20:39:27.000Z",
  "title": "Small ball probabilities for the fractional stochastic heat equation driven by a colored noise",
  "authors": [
    "Jiaming Chen"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2202.04534",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the fractional stochastic heat equation on the $d$-dimensional torus $\\mathbb{T}^d:=[-1,1]^d$, $d\\geq 1$, with periodic boundary conditions: $$ \\partial_t u(t,x)= -(-\\Delta)^{\\alpha/2}u(t,x)+\\sigma(t,x,u)\\dot{F}(t,x)\\quad x\\in \\mathbb{T}^d,t\\in\\mathbb{R}^+ ,$$ where $\\alpha\\in(1,2]$ and $\\dot{F}(t,x)$ is a white in time and colored in space noise. We assume that $\\sigma$ is Lipschitz in $u$ and uniformly bounded. We provide small ball probabilities for the solution $u$ when $u(0,x)\\equiv 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-27T20:39:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional stochastic heat equation driven",
      "small ball probabilities",
      "colored noise",
      "periodic boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}