{
  "id": "2301.03756",
  "version": "v1",
  "published": "2023-01-10T02:21:51.000Z",
  "updated": "2023-01-10T02:21:51.000Z",
  "title": "Brownian Hitting to Spheres",
  "authors": [
    "Yuji Hamana",
    "Hiroyuki Matsumoto"
  ],
  "comment": "13 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $S^{d-1}_r$ be the sphere in $\\bR^d$ whose center is the origin and the radius is $r$, and $\\sigma_r$ be the first hitting time to it of the standard Brownian motion $\\{B_t\\}_{t\\geqq0}$, possibly with constant drift. The aim of this article is to show explicit formulae by means of spherical harmonics for the density of the joint distribution of $(\\sigma_r,B_{\\sigma_r})$ and to study the asymptotic behavior of the distribution function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-10T02:21:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65"
    ],
    "keywords": [
      "brownian hitting",
      "standard brownian motion",
      "distribution function",
      "first hitting time",
      "explicit formulae"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}