{
  "id": "1811.04795",
  "version": "v1",
  "published": "2018-11-12T15:34:51.000Z",
  "updated": "2018-11-12T15:34:51.000Z",
  "title": "On the absolute continuity of random nodal volumes",
  "authors": [
    "Jürgen Angst",
    "Guillaume Poly"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, non-degenerated and stationary Gaussian field $(f(x), {x \\in \\mathbb R^d})$. Under mild conditions, we prove that in dimension $d\\geq 3$, the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable baring in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac--Rice type formulas allowing one to express the volume of the set $\\{f =0\\}$ as integrals of explicit functionals of $(f,\\nabla f,\\text{Hess}(f))$ and next to deduce that the random nodal volume belongs to the domain of a suitable Malliavin gradient. The celebrated Bouleau--Hirsch criterion then gives conditions ensuring the absolute continuity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-12T15:34:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "absolute continuity",
      "random nodal volume belongs",
      "singular part",
      "stationary gaussian field",
      "proving closed kac-rice type formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}