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“Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth
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  • Published: 11 January 2012

“Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth

  • V. Le1,
  • E. Pardoux1 &
  • A. Wakolbinger2 

Probability Theory and Related Fields volume 155, pages 583–619 (2013)Cite this article

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Abstract

We obtain a representation of Feller’s branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion H with a drift that is affine linear in the local time accumulated by H at its current level. As in the classical Ray–Knight representation, the excursions of H are the exploration paths of the trees of descendants of the ancestors at time t = 0, and the local time of H at height t measures the population size at time t. We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time s and living at time t = H s is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating H with a sequence of Harris paths H N which figure in a Ray–Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of H N together with its local times and with the Girsanov densities that introduce the dependence in the reproduction.

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Author information

Authors and Affiliations

  1. Laboratoire d’Analyse, Topologie, Probabilités, Aix-Marseille Université, 13453, Marseille Cedex 13, France

    V. Le & E. Pardoux

  2. Institut für Mathematik, Goethe-Universität Frankfurt, 60325, Frankfurt am Main, Germany

    A. Wakolbinger

Authors
  1. V. Le
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  2. E. Pardoux
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  3. A. Wakolbinger
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Corresponding author

Correspondence to E. Pardoux.

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Le, V., Pardoux, E. & Wakolbinger, A. “Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth. Probab. Theory Relat. Fields 155, 583–619 (2013). https://doi.org/10.1007/s00440-011-0408-x

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  • Received: 05 October 2010

  • Revised: 19 November 2011

  • Published: 11 January 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-011-0408-x

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Keywords

  • Ray–Knight representation
  • Feller branching with logistic growth
  • Exploration process
  • Local time
  • Girsanov transform

Mathematics Subject Classification (2000)

  • Primary 60J70
  • Secondary 60J80
  • 60J55
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