{
  "id": "2203.14394",
  "version": "v1",
  "published": "2022-03-27T21:07:03.000Z",
  "updated": "2022-03-27T21:07:03.000Z",
  "title": "Tightness for Thick Points in two dimensions",
  "authors": [
    "Jay Rosen"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ \\theta$ be the first exit time of the unit disk $D_{1}$. Let \\[\\mu_{ \\theta } ( x,\\epsilon) =\\frac{1}{\\pi\\epsilon^{ 2} }\\int_{0}^{ \\theta }1_{\\{ B( x,\\epsilon)\\}}( W_{t})\\,dt,\\] and set $\\mu^{ \\ast}_{ \\theta } (\\epsilon)=\\sup_{x\\in D_{1}}\\mu_{ \\theta } ( x,\\epsilon)$. We show that \\[\\sqrt{\\mu^{\\ast}_{\\theta} (\\epsilon)}-\\sqrt{2/\\pi} \\left(\\log \\epsilon^{-1}- \\frac{1}{2}\\log\\log \\epsilon^{-1}\\right)\\] is tight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-27T21:07:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65"
    ],
    "keywords": [
      "thick points",
      "dimensions",
      "first exit time",
      "brownian motion",
      "unit disk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}