{
  "id": "2202.04534",
  "version": "v1",
  "published": "2022-02-09T15:58:06.000Z",
  "updated": "2022-02-09T15:58:06.000Z",
  "title": "Small ball probabilities for the stochastic heat equation with colored noise",
  "authors": [
    "Jiaming Chen"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a stochastic heat equation on the 1-dimensional torus $\\mathbb{T}:=[-1,1]$ with periodic boundary conditions: $$\\partial_t u(t,x)=\\partial^2_x u(t,x)+\\sigma(t,x,u)\\dot{F}(t,x),\\quad x\\in \\mathbb{T},t\\in\\mathbb{R}^+$$ and $\\dot{F}(t,x)$ is 2-parameter 1-dimensional white in time, colored in space noise. We assume that $\\sigma$ is Lipschitz in $u$ and uniformly elliptic. We prove small ball probabilities for the solution $u$ when $u(0,x)\\equiv 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-09T15:58:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15"
    ],
    "keywords": [
      "stochastic heat equation",
      "small ball probabilities",
      "colored noise",
      "periodic boundary conditions",
      "space noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}