{
  "id": "0904.4175",
  "version": "v3",
  "published": "2009-04-27T14:44:11.000Z",
  "updated": "2012-06-07T04:38:43.000Z",
  "title": "A continuum-tree-valued Markov process",
  "authors": [
    "Romain Abraham",
    "Jean-François Delmas"
  ],
  "journal": "Annals of Probability 40, 3 (2012) 1167-1211",
  "doi": "10.1214/11-AOP644",
  "categories": [
    "math.PR"
  ],
  "abstract": "We present a construction of a L\\'evy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $\\psi$ be a critical branching mechanism. We set $\\psi_\\theta(\\cdot)=\\psi(\\cdot+\\theta)-\\psi(\\theta)$. Let $\\Theta=(\\theta_\\infty,+\\infty)$ or $\\Theta=[\\theta_\\infty,+\\infty)$ be the set of values of $\\theta$ for which $\\psi_\\theta$ is a branching mechanism. The pruning procedure allows to construct a decreasing L\\'evy-CRT-valued Markov process $(\\ct_\\theta,\\theta\\in\\Theta)$, such that $\\mathcal{T}_\\theta$ has branching mechanism $\\psi_\\theta$. It is sub-critical if $\\theta>0$ and super-critical if $\\theta<0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $\\theta$ for which $\\mathcal{T}_\\theta$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-06-07T04:38:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum-tree-valued markov process",
      "branching mechanism",
      "continuous state branching process",
      "explosion time",
      "levy continuum random tree"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.4175A"
    }
  }
}