{
  "id": "2202.10258",
  "version": "v2",
  "published": "2022-02-21T14:24:03.000Z",
  "updated": "2024-07-04T09:53:42.000Z",
  "title": "Brownian continuum random tree conditioned to be large",
  "authors": [
    "Romain Abraham",
    "Jean-Franç Ois Delmas",
    "Hui He"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a Feller diffusion (Zs, s $\\ge$ 0) (with diffusion coefficient $\\sqrt$ 2$\\beta$ and drift $\\theta$ $\\in$ R) that we condition on {Zt = at}, where at is a deterministic function, and we study the limit in distribution of the conditioned process and of its genealogical tree as t $\\rightarrow$ +$\\infty$. When at does not increase too rapidly, we recover the standard size-biased process (and the associated genealogical tree given by the Kesten's tree). When at behaves as $\\alpha$$\\beta$ 2 t 2 when $\\theta$ = 0 or as $\\alpha$ e 2$\\beta$|$\\theta$|t when $\\theta$ = 0, we obtain a new process whose distribution is described by a Girsanov transformation and equivalently by a SDE with a Poissonian immigration. Its associated genealogical tree is described by an infinite discrete skeleton (which does not satisfy the branching property) decorated with Brownian continuum random trees given by a Poisson point measure. As a by-product of this study, we introduce several sets of trees endowed with a Gromovtype distance which are of independent interest and which allow here to define in a formal and measurable way the decoration of a backbone with a family of continuum random trees.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-07-04T09:53:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian continuum random tree",
      "associated genealogical tree",
      "infinite discrete skeleton",
      "poisson point measure",
      "independent interest"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}