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. 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.
Article PDF
Similar content being viewed by others
References
Ba, M., Pardoux, E., Sow, A.B.: Binary trees, exploration processes, and an extended Ray–Knight Theorem. J. Appl. Probab. 49 (2012)
Billingsley P.: Probability and measure, 3rd edn. Wiley, New York (1995)
Billingsley P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)
Brémaud P.: Point processes and queues: martingale dynamics. Springer-Verlag, New York (1981)
Ethier S., Kurtz Th.: Markov processes: characterization and convergence. Wiley, New York (1986)
Friedman A.: Stochastic differential equations and applications, vol 1. Academic Press, New York (1975)
Joffe A., Métivier M.: Weak convergence of sequences of semi-martingales with applications to multitype branching processes. Adv. Appl. Probab. 18, 20–65 (1986)
Kurtz, Th., Protter, Ph.: Weak convergence of stochastic integrals and differential equations. In: Probabilistic models for nonlinear partial differential equations. Lecture Notes in Math, vol. 1627, pp. 1–41 (1996)
Lambert A.: The branching process with logistic growth. Ann. Appl. Probab. 15, 1506–1535 (2005)
Liptser R.S., Shiryayev A.N.: Theory of martingales. Kluwer Academic Publishers, Dordrecht (1989)
Méléard, S.: Quasi-stationary distributions for population processes. Lecture at CIMPA school, St Louis, Sénégal (2010) http://www.cmi.univ-mrs.fr/~pardoux/Ecole_CIMPA/CoursSMeleard.pdf
Norris J.R., Rogers L.C.G., Williams D.: Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Relat. Fields 74, 271–287 (1987)
Pardoux, E., Wakolbinger, A.: From exploration paths to mass excursions—variations on a theme of Ray and Knight. In: Blath, J., Imkeller, P., Roelly, S. (eds.) Surveys in Stochastic Processes, Proceedings of the 33rd SPA Conference in Berlin, 2009, pp. 87–106. EMS (2011)
Pardoux E., Wakolbinger A.: From Brownian motion with a local time drift to Feller’s branching diffusion with logistic growth. Elec. Commun. Probab. 16, 720–731 (2011)
Perkins E.: Weak invariance principles for local time. Z. Wahrscheinlichkeitstheorie verw. Gebiete 60, 437–451 (1982)
Revuz D., Yor M.: Continuous martingales and Brownian motion, 3rd edn. Spinger Verlag, New York (1999)
Stroock D.W.: Probability theory: an analytic view. Cambridge University Press, Cambridge (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Le, V., Pardoux, E. & Wakolbinger, A. “Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth. Probab. Theory Relat. Fields 155, 583–619 (2013). https://doi.org/10.1007/s00440-011-0408-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-011-0408-x
Keywords
- Ray–Knight representation
- Feller branching with logistic growth
- Exploration process
- Local time
- Girsanov transform