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

Original · EN

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 (see e.g. LG4). 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=Hₛ 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.