{
  "id": "1805.06936",
  "version": "v1",
  "published": "2018-05-17T19:13:09.000Z",
  "updated": "2018-05-17T19:13:09.000Z",
  "title": "Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise",
  "authors": [
    "Raluca M. Balan",
    "Lluís Quer-Sardanyons",
    "Jian Song"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we consider the stochastic wave equation on $\\mathbb{R}_{+} \\times \\mathbb{R}^d$, in spatial dimension $d=1$ or $d=2$, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structure are given by locally integrable functions $\\gamma$ (in time) and $f$ (in space), which are the Fourier transforms of tempered measures $\\nu$ on $\\mathbb{R}$, respectively $\\mu$ on $\\mathbb{R}^d$. Our main result shows that the law of the solution $u(t,x)$ of this equation is absolutely continuous with respect to the Lebesgue measure, provided that the spatial spectral measure $\\mu$ satisfies an integrability condition which ensures that the sample paths of the solution are H\\\"older continuous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-17T19:13:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic wave equation",
      "spatial covariance structure",
      "multiplicative space-time homogeneous gaussian noise",
      "spatial spectral measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}