Abstract
Markovian growth-fragmentation processes introduced by Bertoin extend the pure fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of the known asymptotic behaviors of self-similar pure fragmentations when growth is added to the fragments is a natural question that we investigate in this paper. Our results involve the terminal value of some additive martingales whose uniform integrability is an essential requirement. Dwelling first on the homogeneous case, we exploit the connection with branching random walks and in particular the martingale convergence of Biggins to derive precise asymptotic estimates. The self-similar case is treated in a second part; under the so called Malthusian hypotheses and with the help of several martingale-flavored features recently developed by Bertoin et al., we obtain limit theorems for empirical measures of the fragments.
Highlights
Fragmentation processes are meant to describe the evolution of an object which is subject to random and repeated dislocations over time
The way the mass is spread into smaller fragments during a dislocation event is usually given by a mass-partition, that is an element of the space
With the help of a wellknown theorem due to Biggins [19] and by adapting arguments of Bertoin and Rouault [14], we prove the uniform convergence of additive martingales from which, in the realm of branching random walks, we infer precise estimates for the empirical measure of the fragments and the asymptotic behavior of the largest one
Summary
Fragmentation processes are meant to describe the evolution of an object which is subject to random and repeated dislocations over time. With the help of a wellknown theorem due to Biggins [19] and by adapting arguments of Bertoin and Rouault [14], we prove the uniform convergence of additive martingales from which, in the realm of branching random walks, we infer precise estimates for the empirical measure of the fragments and the asymptotic behavior of the largest one. This part can be viewed as an application to the study of extremal statistics in certain branching random walks, see e.g. the recent developments by Aïdékon [2], Aïdékon et al [1], Arguin et al [3] and Hu et al [25].
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