Asymptotics of self-similar growth-fragmentation processes

Abstract

Markovian growth-fragmentation processes introduced in [8,9] 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 [6,11,12,14] 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 [8], we exploit the connection with branching random walks and in particular the martingale convergence of Biggins [18,19] to derive precise asymptotic estimates. The self-similar case [9] is treated in a second part; under the so called Malthusian hypotheses and with the help of several martingale-flavored features recently developed in [10], we obtain limit theorems for empirical measures of the fragments.

Self-similar solutions of kinetic-type equations: The boundary case

“…The derivation of Proposition 3.2 utilizes ideas borrowed from [18], where part (i) has been stated without a proof in Remark 2.11(iii). Firstly, we obtain two auxiliary lemmas which show that the Biggins martingale (M t (γ * )) t 0 is in the boundary case.…”

Fragmentations with self-similar branching speeds

“…Related and inspiring works include self-similar branching Markov chains [19], the recent so-called branching Lévy processes of [8], as well as many recent developments which have been published on self-similar growth-fragmentation processes (see e.g. [11,14,23]), introduced by Bertoin [7], which allow masses of particles to fluctuate as a positive Markov process. Note the difference between our processes and the so-called (self-similar) multi-type fragmentation processes [6,16,25], for which marks take a finite number of values but determine more than just the speed at which particles undergo fragmentation.…”

The strong Malthusian behavior of growth-fragmentation processes

“…Focusing in a little more detail on the literature on growth-fragmentation processes, Bertoin et al [BBCK18] and Dadoun [Dad17] deal with the self-similar case c(x) = ax α+1 , B(x) = bx α , κ(x, dp) = κ(dp), and look at the convergence (in…”