{
  "id": "2206.02966",
  "version": "v1",
  "published": "2022-06-07T01:58:45.000Z",
  "updated": "2022-06-07T01:58:45.000Z",
  "title": "Inverting Ray-Knight identities on trees",
  "authors": [
    "Xiaodan Li",
    "Yushu Zheng"
  ],
  "comment": "48 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we first introduce the Ray-Knight identity and percolation Ray-Knight identity related to loop soup with intensity $\\alpha (\\ge 0)$ on trees. Then we present the inversions of the above identities, which are expressed in terms of repelling jump processes. In particular, the inversion in the case of $\\alpha=0$ gives the conditional law of a Markov jump process given its local time field. We further show that the fine mesh limits of these repelling jump processes are the self-repelling diffusions \\cite{Aidekon} involved in the inversion of the Ray-Knight identity on the corresponding metric graph. This is a generalization of results in \\cite{2016Inverting,lupu2019inverting,LupuEJP657}, where the authors explore the case of $\\alpha=1/2$ on a general graph. Our construction is different from \\cite{2016Inverting,lupu2019inverting} and based on the link between random networks and loop soups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-07T01:58:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J55",
      "60G55",
      "60J27",
      "60J65"
    ],
    "keywords": [
      "inverting ray-knight identities",
      "repelling jump processes",
      "loop soup",
      "fine mesh limits",
      "markov jump process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}