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Limit theorems for some branching measure-valued processes

Part of: Markov processes Limit theorems Eigenvalue problems

Published online by Cambridge University Press:  26 June 2017

Bertrand Cloez
Bertrand Cloez*
Affiliation:
INRA
*
* Postal address: INRA Montpellier, UMR MISTEA, Bâtiment 29, 2 Place Pierre Viala, 34060 Montpellier Cedex 1, France. Email address: [email protected]
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Abstract

We consider a particle system in continuous time, a discrete population, with spatial motion, and nonlocal branching. The offspring's positions and their number may depend on the mother's position. Our setting captures, for instance, the processes indexed by a Galton–Watson tree. Using a size-biased auxiliary process for the empirical measure, we determine the asymptotic behaviour of the particle system. We also obtain a large population approximation as a weak solution of a growth-fragmentation equation. Several examples illustrate our results. The main one describes the behaviour of a mitosis model; the population is size structured. In this example, the sizes of the cells grow linearly and if a cell dies then it divides into two descendants.

Keywords

Branching Markov processlimit theoremmany-to-one formulasize-biased reproduction distributioneigenproblemdeterministic macroscopic approximationgrowth-fragmentation equation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J75: Jump processes 60F15: Strong theorems 60F05: Central limit and other weak theorems 45C05: Eigenvalue problems 92D25: Population dynamics (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 549 - 580
Copyright
Copyright © Applied Probability Trust 2017 

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