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Regularity of formation of dust in self-similar fragmentations. (English) Zbl 1041.60058
Summary: In self-similar fragmentations with a negative index, fragments split even faster as their mass is smaller, so that the fragmentation runs away and some mass is reduced to dust. Our purpose is to investigate the regularity of this formation of dust. Let \(M(t)\) denote the mass of dust at time \(t\). We give some sufficient and some necessary conditions for the measure \(dM\) to be absolutely continuous. In case of absolute continuity, we obtain an approximation of the density by functions of small fragments. We also study the Hausdorff dimension of \(dM\) and of its support, as well as the Hölder-continuity of the dust’s mass \(M\).

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G17 Sample path properties
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