{
  "id": "1206.1514",
  "version": "v2",
  "published": "2012-06-07T14:46:24.000Z",
  "updated": "2012-06-17T15:52:13.000Z",
  "title": "Champagne subregions with unavoidable bubbles",
  "authors": [
    "Wolfhard Hansen",
    "Ivan Netuka"
  ],
  "comment": "10 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "A champagne subregion of a connected open set $U\\ne\\emptyset$ in $R^d$, $d\\ge 2$, is obtained omitting pairwise disjoint closed balls $\\bar B(x, r_x)$, $x\\in X$, the bubbles, where $X$ is a locally finite set in $U$. The union $A$ of these balls may be unavoidable, that is, Brownian motion, starting in $U\\setminus A$ and killed when leaving $U$, may hit $A$ almost surely or, equivalently, $A$ may have harmonic measure one for $U\\setminus A$. Recent publications by Gardiner/Ghergu ($d\\ge 3$) and by Pres ($d=2$) give rather sharp answers to the question how small such a set $A$ may be, when $U$ is the unit ball. In this paper, using a new criterion for unavoidable sets and a straightforward approach, much stronger results are obtained, results which hold as well for an arbitrary open set $U$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-17T15:52:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31A15",
      "31B15",
      "60J65"
    ],
    "keywords": [
      "champagne subregion",
      "unavoidable bubbles",
      "arbitrary open set",
      "harmonic measure",
      "omitting pairwise disjoint closed balls"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.1514H"
    }
  }
}