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 Electronic Journal of Probability > Vol. 9 (2004) > Paper 4 open journal systems 


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

Bénédicte Haas, Université Pierre et Marie Curie
Grégory Miermont, DMA, Ecole Normale Supérieure, et Université Paris VI


Abstract
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.


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Pages: 57-97

Published on: February 13, 2004
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Electronic Journal of Probability. ISSN: 1083-6489