{
  "id": "1412.1896",
  "version": "v1",
  "published": "2014-12-05T05:27:06.000Z",
  "updated": "2014-12-05T05:27:06.000Z",
  "title": "On Structure of Regular Subspaces of One-dimensional Brownian Motion",
  "authors": [
    "Liping Li",
    "Jiangang Ying"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "The main purpose of this paper is to explore the structure of regular subspaces of 1-dim Brownian motion. As outlined in \\cite{FMG} every such regular subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^c$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular subspace on $F$ is still a regular subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover we have proved that the trace of Brownian motion on $F$ may be decomposed into two part, one is the trace of the regular subspace on $F$, which has only the non-local part and the other comes from the orthogonal complement of the regular subspace, which has only the local part. Actually the orthogonal complement of regular subspace corresponds to a time-changed Brownian motion after a darning transform.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-05T05:27:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C25",
      "60J55",
      "60J60"
    ],
    "keywords": [
      "one-dimensional brownian motion",
      "trace dirichlet form",
      "orthogonal complement",
      "regular subspace corresponds",
      "inclusion relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.1896L"
    }
  }
}