Electronic Journal of Probability

The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree

Bénédicte Haas and Grégory Miermont

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We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Continuum Random Tree is encoded in a normalized Brownian excursion. Under mild hypotheses, we then compute the Hausdorff dimensions of these trees, and the maximal Hölder exponents of the height functions.

Article information

Electron. J. Probab. Volume 9 (2004), paper no. 4, 57-97.

Accepted: 9 February 2004
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G18: Self-similar processes 60J25: Continuous-time Markov processes on general state spaces 60G09: Exchangeability

This work is licensed under a Creative Commons Attribution 3.0 License.


Haas, Bénédicte; Miermont, Grégory. The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree. Electron. J. Probab. 9 (2004), paper no. 4, 57--97. doi:10.1214/EJP.v9-187. https://projecteuclid.org/euclid.ejp/1465229690

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