research

Trees under attack: a Ray-Knight representation of Feller's branching diffusion with logistic growth

Abstract

We obtain a representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion HH with a drift that is affine linear in the local time accumulated by HH at its current level. As in the classical Ray-Knight representation, the excursions of HH are the exploration paths of the trees of descendants of the ancestors at time t=0t=0, and the local time of HH at height tt measures the population size at time tt (see e.g. \cite{LG4}). We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time ss and living at time t=Hst=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 HH with a sequence of Harris paths HNH^N which figure in a Ray-Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of HNH^N together with its local times {\em and} with the Girsanov densities that introduce the dependence in the reproduction

    Similar works

    Full text

    thumbnail-image

    Available Versions

    Last time updated on 11/11/2016