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

• 成果类型: Article
• 署名作者: Le, V.; Pardoux, E.; Wakolbinger, A.
• 署名单位: Aix-Marseille Universite; Goethe University Frankfurt
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0408-x
• 发表日期: 2013
• 页码: 583-619
• 关键词: brownian-motion

摘要

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.