{
  "id": "math/0606751",
  "version": "v1",
  "published": "2006-06-29T10:53:45.000Z",
  "updated": "2006-06-29T10:53:45.000Z",
  "title": "On the transience of processes defined on Galton--Watson trees",
  "authors": [
    "Andrea Collevecchio"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117905000000837 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 3, 870-878",
  "doi": "10.1214/009117905000000837",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce a simple technique for proving the transience of certain processes defined on the random tree $\\mathcal{G}$ generated by a supercritical branching process. We prove the transience for once-reinforced random walks on $\\mathcal{G}$, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567--592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on $\\mathcal{G}$. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229--1241] and [Ann. Probab. 18 (1990) 931--958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42--62] proved that a vertex-reinforced jump process defined on the $b$-ary tree is transient if $b\\ge 4$ and recurrent if $b=1$. The case $b=2$ is still open.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-29T10:53:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60J80",
      "60J75"
    ],
    "keywords": [
      "galton-watson trees",
      "transience",
      "vertex-reinforced jump process",
      "theory related fields"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6751C"
    }
  }
}