Abstract.
We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Lévy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin.
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Mathematics Subject Classification (2000): 60J25, 60G52
Acknowledgement Many thanks to Jean Bertoin for many precious comments on this work, and to Jean-François Le Gall for discussions related to the stable tree. Thanks also to an anonymous referee for a careful reading and very helpful comments that helped to consequently improve the presentation of this work.
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Miermont, G. Self-similar fragmentations derived from the stable tree II: splitting at nodes. Probab. Theory Relat. Fields 131, 341–375 (2005). https://doi.org/10.1007/s00440-004-0373-8
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DOI: https://doi.org/10.1007/s00440-004-0373-8