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

2004/01/01 by Bénédicte Haas, Grégory Miermont · 4 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Financial Risk and Volatility Modeling #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.1214/ejp.v9-187

openalex publication_date 2004/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

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.

Citations

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

Cited by

• A new family of Markov branching trees: the alpha-gamma model
• Fragmentations with self-similar branching speeds
• Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
• Grégory Miermont [wikipedia]