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Self-similar fragmentations. (English) Zbl 1002.60072
Author’s summary: We introduce a probabilistic model that meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such process is determined by its index of self-similarity \(\alpha\in {\mathbf R}\), a rate of erosion \(c\geq 0\), and a so-called Lévy measure that accounts for sudden dislocations. The key of the analysis is provided by a transformation of self-similar fragmentations which enables us to reduce the study to the homogeneous case \(\alpha=0\).
For related papers see D. J. Aldous [Bernoulli 5, No. 1, 3-48 (1999; Zbl 0930.60096)] and the author [Probab. Theory Relat. Fields 117, No. 2, 289-301 (2000; Zbl 0965.60072)].

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G09 Exchangeability for stochastic processes
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