{
  "id": "1812.09949",
  "version": "v1",
  "published": "2018-12-24T16:28:29.000Z",
  "updated": "2018-12-24T16:28:29.000Z",
  "title": "Fréchet differentiability of mild solutions to SPDEs with respect to the initial datum",
  "authors": [
    "Carlo Marinelli",
    "Luca Scarpa"
  ],
  "comment": "27 pages, no figures",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We establish n-th order Fr\\'echet differentiability with respect to the initial datum of mild solutions to a class of jump-diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order G\\^ateaux differentiability of their solutions with respect to the initial datum, extending previous results in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-24T16:28:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "initial datum",
      "mild solutions",
      "fréchet differentiability",
      "establish n-th order frechet differentiability",
      "non-local kolmogorov equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}