Open Collections

Description

Growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. In the self-similar case, it is known that a simple Malthusian condition ensures that 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. We shall present here the converse: when this Malthusian 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. Based on a joint work with Robin Stephenson.