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

• Published in: Probability theory and related fields 2013-04, Vol.155 (3-4), p.583-619
• Main Authors: Le, V., Pardoux, E., Wakolbinger, A.
• Format: Article
• Language: English
• Subjects: Analysis; Approximation; Brownian motion; Convergence; Density; Diffusion; Economics; Finance; Insurance; Logistics; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Population; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Representations; Reproduction; Statistics for Business; Studies; Theoretical; Trees
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-011-0408-x

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.