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

TL;DR

This paper presents a novel Ray-Knight type representation of Feller's branching diffusion with logistic growth using local times of a reflected Brownian motion with a specific drift, capturing dependence in reproduction.

Contribution

It introduces a new representation linking logistic growth in branching processes to local times of a reflected Brownian motion with drift, accounting for dependence among individuals.

Findings

Representation of logistic branching diffusion via local times of reflected Brownian motion.

Introduction of a pecking order to model dependence in reproduction.

Convergence results connecting Harris paths with the branching process.

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 $t$ (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 $s$ 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 {\em and} with the Girsanov densities that introduce the dependence in the reproduction.