{
  "id": "1702.08183",
  "version": "v1",
  "published": "2017-02-27T08:29:04.000Z",
  "updated": "2017-02-27T08:29:04.000Z",
  "title": "Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet",
  "authors": [
    "Robert C. Dalang",
    "T. Mountford"
  ],
  "comment": "145 pages, 12 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We first consider the additive Brownian motion process $(X(s_1,s_2),\\ (s_1,s_2) \\in \\mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set $\\{(s_1,s_2)\\in \\mathbb{R}^2: X(s_1,s_2) >0\\}$ is equal to $$ \\frac{1}{4}\\left(1 + \\sqrt{13 + 4 \\sqrt{5}}\\right) \\simeq 1.421\\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-27T08:29:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G60",
      "60G17",
      "60G15"
    ],
    "keywords": [
      "hausdorff dimension",
      "standard brownian sheet",
      "additive brownian motion process",
      "random set",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 145,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}