Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Branching processes with interaction and a generalized Ray–Knight Theorem

Mamadou Ba and Etienne Pardoux

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Abstract

We consider a discrete model of population dynamics with interaction between individuals, where the birth and death rates are nonlinear functions of the population size. We obtain the large population limit of a renormalization of our model as the solution of the SDE

\[Z^{x}_{t}=x+\int_{0}^{t}f(Z^{x}_{s})\,\mathrm{d}s+2\int_{0}^{t}\int_{0}^{Z^{x}_{s}}W(\mathrm{d}s,\mathrm{d}u),\] where $W(\mathrm{d}s,\mathrm{d}u)$ is a time space white noise on $[0,\infty)^{2}$.

We give a Ray–Knight representation of this diffusion in terms of the local times of a reflected Brownian motion $H$ with a drift that depends upon the local time accumulated by $H$ at its current level, through the function $f'/2$.

Résumé

Nous considérons un modèle d’évolution d’une population avec interaction entre les individus, où les taux de naissance et de mort sont fonction de la taille de la population. Nous obtenons la limite en grande population après renormalisation, qui est solution de l’EDS

\[Z^{x}_{t}=x+\int_{0}^{t}f(Z^{x}_{s})\,\mathrm{d}s+2\int_{0}^{t}\int_{0}^{Z^{x}_{s}}W(\mathrm{d}s,\mathrm{d}u),\] où $W(\mathrm{d}s,\mathrm{d}u)$ est un bruit blanc sur $[0,\infty)^{2}$.

Nous donnons une représentation de cette diffusion à la Ray–Knight, en fonction des temps locaux d’un mouvement brownien réfléchi $H$ avec une dérive qui dépend du temps local accumulé par $H$ à son niveau courant, à travers la fonction $f'/2$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1290-1313.

Dates
Received: 3 April 2013
Revised: 22 April 2014
Accepted: 22 April 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432043

Digital Object Identifier
doi:10.1214/14-AIHP621

Mathematical Reviews number (MathSciNet)
MR3414448

Zentralblatt MATH identifier
1329.60298

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles
Secondary: 92D25: Population dynamics (general)

Keywords
Galton–Watson processes with interaction Generalized Feller diffusion

Citation

Ba, Mamadou; Pardoux, Etienne. Branching processes with interaction and a generalized Ray–Knight Theorem. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1290--1313. doi:10.1214/14-AIHP621. https://projecteuclid.org/euclid.aihp/1445432043


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