The genealogy of self-similar fragmentations with negative index as a continuum random tree. (English) Zbl 1064.60076

Summary: 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.


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