{
  "id": "2401.15427",
  "version": "v1",
  "published": "2024-01-27T14:40:48.000Z",
  "updated": "2024-01-27T14:40:48.000Z",
  "title": "A regularity property of fractional Brownian sheets",
  "authors": [
    "Philippe Bouafia",
    "Thierry De Pauw"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "A function $f$ defined on $[0, 1]^d$ is called strongly chargeable if there is a continuous vector-field $v$ such that $f(x_1, \\dots,x_d)$ equals the flux of $v$ through the rectangle $[0, x_1] \\times \\cdots \\times [0, x_d]$ for all $(x_1, \\dots, x_d) \\in [0, 1]^d$. In other words, $f$ is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with $d \\geq 2$ parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter $(H_1, \\dots, H_d)$ are shown to be almost surely strongly chargeable whenever \\[ \\frac{H_1 + \\cdots + H_d}{d} > \\frac{d - 1}{d}. \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-27T14:40:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G22",
      "60G17",
      "26A45"
    ],
    "keywords": [
      "fractional brownian sheet",
      "regularity property",
      "continuous vector-field",
      "strongly chargeable",
      "hurst parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}