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Self-similar fragmentations derived from the stable tree I

Abstract

The basic object we consider is a certain model of continuum random tree, called the stable tree. We construct a fragmentation process (F (t),t≥0) out of this tree by removing the vertices located under height t. Thanks to a self-similarity property of the stable tree, we show that the fragmentation process is also self-similar. The semigroup and other features of the fragmentation are given explicitly. Asymptotic results are given, as well as a couple of related results on continuous-state branching processes.

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Author information

Correspondence to Grégory Miermont.

Additional information

Research supported in part by NSF Grant DMS-0071448.

Mathematics Subject Classification (2000): 60J25, 60G52, 60J80

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Miermont, G. Self-similar fragmentations derived from the stable tree I. Probab. Theory Relat. Fields 127, 423–454 (2003). https://doi.org/10.1007/s00440-003-0295-x

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Keywords

  • Self-similar fragmentation
  • stable tree
  • stable process
  • continuous state branching process