{
  "id": "1606.00630",
  "version": "v1",
  "published": "2016-06-02T11:31:11.000Z",
  "updated": "2016-06-02T11:31:11.000Z",
  "title": "Regular Dirichlet extensions of one-dimensional Brownian motion",
  "authors": [
    "Liping Li",
    "Jiangang Ying"
  ],
  "comment": "29 pages with 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "The regular Dirichlet extension is the dual concept of regular Dirichlet subspace. The main purpose of this paper is to characterize all the regular Dirichlet extensions of one-dimensional Brownian motion and to explore their structures. It is shown that every regular Dirichlet extension of one-dimensional Brownian motion may essentially decomposed into at most countable disjoint invariant intervals and an $\\mathcal{E}$-polar set relative to this regular Dirichlet extension. On each invariant interval the regular Dirichlet extension is characterized uniquely by a scale function in a given class. To explore the structure of regular Dirichlet extension we apply the idea introduced in [17], we formulate the trace Dirichlet forms and attain the darning process associated with the restriction to each invariant interval of the orthogonal complement of $H^1_\\mathrm{e}(\\mathbb{R})$ in the extended Dirichlet space of the regular Dirichlet extension. As a result, we find an answer to a long-standing problem whether a pure jump Dirichlet form has proper regular Dirichlet subspaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-02T11:31:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C25",
      "60J55",
      "60J60"
    ],
    "keywords": [
      "regular dirichlet extension",
      "one-dimensional brownian motion",
      "proper regular dirichlet subspaces",
      "pure jump dirichlet form",
      "countable disjoint invariant intervals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}