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 Holder exponents of the height functions.
Highlights
Self-similar fragmentation processes describe the evolution of an object that falls apart, so that different fragments keep on collapsing independently with a rate that depends on their sizes to a certain power, called the index of the self-similar fragmentation
A genealogy is naturally associated to such fragmentation processes, by saying that the common ancestor of two fragments is the block that included these fragments for the last time, before a dislocation had definitely separated them
It turns out that trees have played a key role in models involving self-similar fragmentations, notably, Aldous and Pitman [3] have introduced a way to log the so-called Brownian Continuum Random Tree (CRT) [2] that is related to the standard additive coalescent
Summary
Self-similar fragmentation processes describe the evolution of an object that falls apart, so that different fragments keep on collapsing independently with a rate that depends on their sizes to a certain power, called the index of the self-similar fragmentation. There exists a continuous random function (HF (s), 0 ≤ s ≤ 1), called the height function, such that HF (0) = HF (1), HF (s) > 0 for every s ∈ (0, 1), and such that F has the same law as the fragmentation F ′ defined by: F ′(t) is the decreasing rearrangement of the lengths of the interval components of the open set IF (t) = {s ∈ (0, 1) : HF (s) > t}. The Hausdorff dimension of TF is in general not equal to the inverse of the maximal Holder coefficient of the height process, as one could have expected This turns out to be true in the case of the stable tree, as will be checked in Section 4.4: Corollary 3. If lim supx→0 x−aς(x) < ∞ for some a < 0 and S s−1 1 − 1 ν(ds) < ∞, the Hausdorff dimension is larger than 1/ |a| and the height function cannot have a Holder coefficient γ > θsup ∧ |a|
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