Abstract
We consider a self-similar fragmentation process in which the generic particle of mass $x$ is replaced by the offspring particles at probability rate $x^\alpha$, with positive parameter $\alpha$. The total of offspring masses may be both larger or smaller than $x$ with positive probability. We show that under certain conditions the typical mass in the ensemble is of the order $t^{-1/\alpha}$ and that the empirical distribution of masses converges to a random limit which we characterise in terms of the reproduction law.
Highlights
We study the following continuous-time model of particle fragmentation
During the life-time the mass does not vary and at the end the particle splits into fragments of masses xξj, where {ξj} is independent of the lifetime of the particle and follows a given probability distribution called reproduction law
Conservative or dissipative fragmentations can be treated both as continuous-time interval splitting schemes, similar to discrete-time random recursive constructions, or as state-discretised processes with values in Kingman’s partition structures [5, 6, 7, 8, 10]. These approaches fail completely if the reproduction law allows the possibility of mass creation, when the total mass of the offspring may exceed the mass of the parent particle
Summary
We study the following continuous-time model of particle fragmentation. Each particle in ensemble is characterised by a positive quantity which we call mass. Conservative or dissipative fragmentations can be treated both as continuous-time interval splitting schemes, similar to discrete-time random recursive constructions (as in [23]), or as state-discretised processes with values in Kingman’s partition structures [5, 6, 7, 8, 10] These approaches fail completely if the reproduction law allows the possibility of mass creation, when the total mass of the offspring may exceed the mass of the parent particle. By allocating particle of mass x at − log x the fragmentation process can be seen as a branching random walk with location-dependent sojourn times From this viewpoint, a constraint on the sum of masses seems rather odd, which suggests that such a condition is not essential for the asymptotics.
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