{
  "id": "2206.11361",
  "version": "v1",
  "published": "2022-06-22T20:19:11.000Z",
  "updated": "2022-06-22T20:19:11.000Z",
  "title": "Parabolic Anderson model with rough noise in space and rough initial conditions",
  "authors": [
    "Raluca M. Balan",
    "Le Chen",
    "Yiping Ma"
  ],
  "comment": "13 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this note, we consider the parabolic Anderson model on $\\mathbb{R}_{+} \\times \\mathbb{R}$, driven by a Gaussian noise which is fractional in time with index $H_0>1/2$ and fractional in space with index $0<H<1/2$ such that $H_0+H>3/4$. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all $p$-th moments with $p\\ge 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-22T20:19:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic anderson model",
      "rough initial conditions",
      "rough noise",
      "exponential upper bounds",
      "gaussian noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}