Local explosion in self-similar growth-fragmentation processes

TL;DR

This paper investigates the conditions under which self-similar growth-fragmentation processes experience local explosion, establishing that failure of a Malthusian condition leads to almost sure explosion, using martingale and spine decomposition techniques.

Contribution

It proves that when the Malthusian condition is not met, self-similar growth-fragmentation processes almost surely explode, providing a converse to previous non-explosion results.

Findings

Failure of Malthusian condition causes explosion

Use of additive martingale and spine decomposition

Almost sure explosion when condition not verified

Abstract

Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in a work of Bertoin, where special attention was given to the self-similar case. A Malthusian condition was notably given under which the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. Our main result in this work states the converse: when this condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes.