Homogeneous multitype fragmentations. (English) Zbl 1153.60046

Bertoin, Jean

Sidoravicius, Vladas (ed.) et al., In and out of equilibrium 2. Papers celebrating the 10th edition of the Brazilian school of probability (EBP), Rio de Janiero, Brazil, July 30 to August 4, 2006. Basel: Birkhäuser (ISBN 978-3-7643-8785-3/hbk). Progress in Probability 60, 161-183 (2008).

A homogeneous mass-fragmentation, as it has been defined by J. Bertoin [Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics 102. Cambridge: Cambridge University Press (2006; Zbl 1107.60002)], describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a type may represent, for instance, a geometrical shape, and can take finitely many values. In this setting, the dynamics of a randomly tagged fragment play a crucial role in the analysis of the fragmentation. They are determined by a Markov additive process whose distribution depends explicitly on the characteristics of the fragmentation. As applications, we make the connection with multitype branching random walks explicit, and obtain multitype analogs of the pathwise central limit theorem and large deviation estimates for the empirical distribution of fragments.
For the entire collection see [Zbl 1141.82002].

Reviewer: Valentin Topchii (Omsk)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G18	Self-similar stochastic processes

Keywords:

multitype; fragmentation; branching process; Markov additive process

Citations:

Zbl 1107.60002