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Growth of Lévy trees
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  • Published: 06 March 2007

Growth of Lévy trees

  • Thomas Duquesne1 &
  • Matthias Winkel2 

Probability Theory and Related Fields volume 139, pages 313–371 (2007)Cite this article

Abstract

We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuous-state branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of a Poisson sampling directed by the mass measure.

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Author information

Authors and Affiliations

  1. Département de Mathématiques, Université Paris 11, 91405, Orsay Cedex, France

    Thomas Duquesne

  2. Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK

    Matthias Winkel

Authors
  1. Thomas Duquesne
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  2. Matthias Winkel
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Corresponding author

Correspondence to Matthias Winkel.

Additional information

T. Duquesne is supported by NSF Grants DMS-0203066 and DMS-0405779. M. Winkel is supported by Aon and the Institute of Actuaries, EPSRC Grant GR/T26368/01, le département de mathématique de l’Université d’Orsay and NSF Grant DMS-0405779.

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Duquesne, T., Winkel, M. Growth of Lévy trees. Probab. Theory Relat. Fields 139, 313–371 (2007). https://doi.org/10.1007/s00440-007-0064-3

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  • Received: 24 September 2005

  • Revised: 13 September 2006

  • Published: 06 March 2007

  • Issue Date: November 2007

  • DOI: https://doi.org/10.1007/s00440-007-0064-3

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Keywords

  • Tree-valued Markov process
  • Galton–Watson branching process
  • Genealogy
  • Continuous-state branching process
  • Percolation
  • Gromov–Hausdorff topology
  • Continuum random tree
  • Edge lengths
  • Real tree

Mathematics Subject Classification (2000)

  • 60J80
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