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Asymptotic laws for nonconservative self-similar fragmentations. (English) Zbl 1064.60075

Summary: 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.

MSC:

60G18 Self-similar stochastic processes
60J25 Continuous-time Markov processes on general state spaces

Keywords:

power laws
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