Probability Theory and Related Fields

, Volume 155, Issue 3, pp 583–619

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

Article

DOI: 10.1007/s00440-011-0408-x

Cite this article as:
Le, V., Pardoux, E. & Wakolbinger, A. Probab. Theory Relat. Fields (2013) 155: 583. doi:10.1007/s00440-011-0408-x

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 tHs 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 HN which figure in a Ray–Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of HN together with its local times and with the Girsanov densities that introduce the dependence in the reproduction.

Keywords

Ray–Knight representationFeller branching with logistic growthExploration processLocal timeGirsanov transform

Mathematics Subject Classification (2000)

Primary 60J70Secondary 60J8060J55

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Laboratoire d’Analyse, Topologie, ProbabilitésAix-Marseille UniversitéMarseille Cedex 13France
  2. 2.Institut für MathematikGoethe-Universität FrankfurtFrankfurt am MainGermany