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 [3], 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.
Highlights
A growth-fragmentation process can be viewed as a branching particle system, in which each particle has a mass that evolves continuously as time passes, independently of the other particles, and splits in two
We look at X(t) as the mass of a particle at time t and whenever X makes a jump, we consider this as giving birth to a new particle, whose original mass is equal to the size of the jump
One of the main results established in [3] is that there is a simple Malthusian condition, which is given in terms of the characteristics of the self-similar Markov process X, that ensures that a.s., for all times t 0, the particles generated by the growth-fragmentation can be listed in the non-increasing order of their masses and form a null sequence
Summary
A growth-fragmentation process can be viewed as a branching particle system, in which each particle has a mass that evolves continuously (and in particular may grow) as time passes, independently of the other particles, and splits in two. One of the main results established in [3] is that there is a simple Malthusian condition, which is given in terms of the characteristics of the self-similar Markov process X, that ensures that a.s., for all times t 0, the particles generated by the growth-fragmentation can be listed in the non-increasing order of their masses and form a null sequence. We give a precise construction of the growth-fragmentation process, first assuming that the total mass of Λ1 is finite and treating the general case. This construction is reminiscent of the “branching Lévy process” from [2]. We introduce some important notions and notation, and state our main theorem
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