{
  "id": "1005.5275",
  "version": "v1",
  "published": "2010-05-28T12:09:24.000Z",
  "updated": "2010-05-28T12:09:24.000Z",
  "title": "The Stochastic Wave Equation with Multiplicative Fractional Noise: a Malliavin calculus approach",
  "authors": [
    "Raluca M. Balan"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index $H>1/2$, and has a homogeneous spatial covariance structure given by the Riesz kernel of order $\\alpha$. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is $\\alpha>d-2$, which coincides with the condition obtained in Dalang (1999), when the noise is white in time. Under this condition, we obtain estimates for the $p$-th moments of the solution, we deduce its H\\\"older continuity, and we show that the solution is Malliavin differentiable of any order. When $d \\leq 2$, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-28T12:09:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H15",
      "60H07"
    ],
    "keywords": [
      "stochastic wave equation",
      "malliavin calculus approach",
      "multiplicative fractional noise",
      "homogeneous spatial covariance structure",
      "integral equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.5275B"
    }
  }
}