{
  "id": "1611.06782",
  "version": "v1",
  "published": "2016-11-21T13:38:43.000Z",
  "updated": "2016-11-21T13:38:43.000Z",
  "title": "The orthogonal complements of $H^1(\\mathbb{R})$ in its regular Dirichlet extensions",
  "authors": [
    "Yuncong Shen",
    "Liping Li",
    "Jiangang Ying"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the regular Dirichlet extension $(\\mathcal{E},\\mathcal{F})$ for one-dimensional Brownian motion, that $H^1(\\mathbb{R})$ is a subspace of $\\mathcal{F}$ and $\\mathcal{E}(f,g)=\\frac12\\mathbf{D}(f,g)$ for $f,g\\in H^1(\\mathbb{R})$. Both $H^1(\\mathbb{R})$ and $\\mathcal{F}$ are Hilbert spaces under $\\mathcal{E}_\\alpha$ and hence there is $\\alpha$-orthogonal compliment $\\mathcal{G}_\\alpha$. We give the explicit expression for functions in $\\mathcal{G}_\\alpha$ which then can be described by another two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the darning method, their regular representations are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-21T13:38:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C25",
      "60J55",
      "60J60"
    ],
    "keywords": [
      "regular dirichlet extension",
      "orthogonal complements",
      "natural dirichlet form",
      "one-dimensional brownian motion",
      "hilbert spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}