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Self-similar fragmentations derived from the stable tree II: splitting at nodes
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  • Published: 05 July 2004

Self-similar fragmentations derived from the stable tree II: splitting at nodes

  • Grégory Miermont1 

Probability Theory and Related Fields volume 131, pages 341–375 (2005)Cite this article

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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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Authors and Affiliations

  1. DMA, École Normale Supérieure, and LPMA, Université Paris VI, 45, rue d’Ulm, 75230, Paris Cedex 05, France

    Grégory Miermont

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  1. Grégory Miermont
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Correspondence to Grégory Miermont.

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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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  • Received: 04 July 2003

  • Revised: 05 May 2004

  • Published: 05 July 2004

  • Issue Date: March 2005

  • DOI: https://doi.org/10.1007/s00440-004-0373-8

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Key words or phrases:

  • Self-similar fragmentation
  • Stable tree
  • Stable processes
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