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Fragmentations with self-similar branching speeds

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  22 November 2021

Jean-Jil Duchamps Open the ORCID record for Jean-Jil Duchamps [Opens in a new window]
Jean-Jil Duchamps*
Affiliation:
Université Bourgogne Franche-Comté
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Abstract

We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

Keywords

Self-similarbranching processexchangeablefragmentationrandom partitionrandom treeLévy process

MSC classification

Primary: 60G18: Self-similar processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G09: Exchangeability 60G51: Processes with independent increments; L'evy processes 60G25: Prediction theory
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 1149 - 1189
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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Footnotes

*

Postal address: Laboratoire de mathématiques de Besançon, UMR 6623, CNRS, Université Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon CEDEX, France.

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