{
  "id": "1506.02021",
  "version": "v1",
  "published": "2015-06-05T19:37:11.000Z",
  "updated": "2015-06-05T19:37:11.000Z",
  "title": "The spans in Brownian motion",
  "authors": [
    "Steven N. Evans",
    "Jim Pitman",
    "Wenpin Tang"
  ],
  "comment": "33 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "For $d \\in \\{1,2,3\\}$, let $(B^d_t;~ t \\geq 0)$ be a $d$-dimensional standard Brownian motion. We study the $d$-Brownian span set $Span(d):=\\{t-s;~ B^d_s=B^d_t~\\mbox{for some}~0 \\leq s \\leq t\\}$. We prove that almost surely the random set $Span(d)$ is $\\sigma$-compact and dense in $\\mathbb{R}_{+}$. In addition, we show that $Span(1)=\\mathbb{R}_{+}$ almost surely; the Lebesgue measure of $Span(2)$ is $0$ almost surely and its Hausdorff dimension is $1$ almost surely; and the Hausdorff dimension of $Span(3)$ is $\\frac{1}{2}$ almost surely. We also list a number of conjectures and open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-05T19:37:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "60J65"
    ],
    "keywords": [
      "dimensional standard brownian motion",
      "hausdorff dimension",
      "brownian span set",
      "random set",
      "lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150602021E"
    }
  }
}