{
  "id": "2005.01370",
  "version": "v1",
  "published": "2020-05-04T10:32:30.000Z",
  "updated": "2020-05-04T10:32:30.000Z",
  "title": "A nonlinear Schr{ö}dinger equation with fractional noise",
  "authors": [
    "Aurélien Deya",
    "Nicolas Schaeffer",
    "Laurent Thomann"
  ],
  "comment": "43 pages",
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "We study a stochastic Schr{\\\"o}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schr{\\\"o}dinger group, which allows us to cope with the strong irregularity of the solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-04T10:32:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dinger equation",
      "fractional noise",
      "nonlinear schr",
      "space-time fractional perturbation",
      "small hurst index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}