Asymptotics of self-similar growth-fragmentation processes

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.

arXiv comment33 pages; published in EJP

References in corpus (5)

• Convergence in law of the minimum of a branching random walk
• Exponential functionals of Levy processes
• A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes
• Law of the absorption time of some positive self-similar Markov processes
• Asymptotical behaviour of the presence probability in branching random walks and fragmentations

Cited by in corpus (8)

• The strong Malthusian behavior of growth-fragmentation processes
• Biggins' Martingale Convergence for Branching Lévy Processes
• Probability tilting of compensated fragmentations
• Growth-fragmentation processes and bifurcators
• Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes
• A growth-fragmentation model related to Ornstein-Uhlenbeck type processes
• Solutions of kinetic-type equations with perturbed collisions
• Fragmentations with self-similar branching speeds