{
  "id": "1509.06616",
  "version": "v1",
  "published": "2015-09-22T14:18:15.000Z",
  "updated": "2015-09-22T14:18:15.000Z",
  "title": "Excursion theory for Brownian motion indexed by the Brownian tree",
  "authors": [
    "Céline Abraham",
    "Jean-François Le Gall"
  ],
  "comment": "43 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical It\\^o theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connectedcomponent is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism $\\psi(u)=\\sqrt{8/3}\\,u^{3/2}$. Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure $\\mathbb{M}_0$ which is the analog of the It\\^o measure of Brownian excursions. We provide various descriptions of the excursion measure $\\mathbb{M}_0$, and we also determine several explicit distributions, such as the joint distribution of the boundary length and the mass of an excursion under $\\mathbb{M}_0$. We use the Brownian snake as a convenient tool for defining and analysing the excursions of our tree-indexed Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-22T14:18:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J68",
      "60J80",
      "60J65"
    ],
    "keywords": [
      "excursion theory",
      "brownian tree",
      "tree-indexed brownian motion",
      "excursion measure",
      "boundary lengths coincides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}