Self-similar fragmentations

By Jean Bertoin. 2002

183 citations recorded by the source corpus.

Subfields: Condensed Matter Physics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9d8c986d721d612039455a742291ef1041057ba9

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In Condensed Matter Physics

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Citations in the corpus

A continuum-tree-valued Markov process (2009, unknown): “We thus recover a well-known result from Aldous and Pitman [10] on the size process of a tagged fragment for a self-similar fragmentation (see [14]) with index 1/2, no erosion and binary dislocation measure ν defined on pairs (s1, s2) such that s1 ≥ s2 ≥ 0 and s1 + s2 = 1 by ν(s1 ∈ dx) = (2πx(3)(1− ”
The asymptotic behavior of fragmentation processes (2003, methodology): “In [6,7], we considered a class of Markov processes X = (X(t), t ≥ 0) with values in S↓, called self-similar fragmentations, which fulfill both the so-called fragmentation and the scaling properties.”
Conditioned stable Levy processes and Lamperti representation. (2006, unknown): “Note also that the constant a is related to c + , c − and α as follows: a = c + − c − 1 − α , α 6 = 1.”
The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree (2004, unknown): “According to [7], the fragmentation property holds actually for G -stopping times and we shall refer to it as the strong fragmentation property.”
A new family of Markov branching trees: the alpha-gamma model (2008, background): “Bertoin [5] proved that the distribution of a P N -valued self-similar fragmentation process is determined by a triple ( a, c, ν ), where a ∈ R , c ≥ 0 and ν is a dislocation measure on S ↓ .”
Homogeneous fragmentation processes (2001, background): “In a related work [ 6 ], it is shown that the more general class of so-called selfsimilar fragmentation processes (which includes the fragmentation considered by Aldous and Pitman [3] and Bertoin [5]) can be constructed from homogeneous fragmentation processes via some time-substitution which enable”
Spinal partitions and invariance under re-rooting of continuum random trees (2007, background): “For details on their fragmentation properties, see Bertoin [9] for the Brownian case (i.”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, background): “Haas and Miermont [17] and Bertoin [3] observed the following.”
Self-similar fragmentations derived from the stable tree I (2003, methodology): “Surprisingly, although this other device looks quite different from the first (no mass is lost when cutting hubs, whereas there is a loss of mass when we throw everything that is located under the height h), it turns out that the only difference between these two fragmentations is the speed at which”
Random Recursive Trees and the Bolthausen-Sznitman Coalesent (2005, result): “We note that p̂1,2 6= p̂2,3 and so the time-reversal of the Bolthausen-Sznitman coalescent is not self-similar in the sense of Bertoin [3].”
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (2010, background): “Bertoin [8] and Berestycki [6] have shown that the laws of self-similar fragmentation processes are characterized by three parameters: the index a, a non-negative erosion coefficient, and a dislocation measure ν on S↓.”
Combinatorial Stochastic Processes (2006, methodology): “A natural guess is that such a construction might be made with one of the self-similar fragmentation processes of Bertoin [46], but Miermont and Schweinsberg [312] have shown that a construction of this form is possible only for α = 12 .”
Law of the absorption time of some positive self-similar Markov processes (2009, background): “It is then not very surprising that these processes appear in different settings, such as, for instance, coagulation-fragmentation processes [5], and continuous state branching processes with immigration [35].”
Discretization Methods for Homogeneous Fragmentations (2004, methodology): “In this direction, it is convenient to use an interval representation of the fragmentation (see [5]).”
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models (2006, methodology): “Lemma 5 ([8]) Let (Π0(t), t ≥ 0) be a standard homogeneous fragmentation with dislocation measure ν, and let a ∈ R.”
Self-similar fragmentations derived from the stable tree II: splitting at nodes (2004, methodology): “In [9], Bertoin showed that the erosion of FAP is 0, and that the dislocation measure νAP is characterized by the two formulas νAP(s1 ∈ dx) = dx √ 2πx3(1 − x)3 , x ∈ [1/2, 1),”
Poisson-Kingman partitions (2002, background): “A natural guess is that such a construction might be made with one of the self-similar fragmentation processes of Bertoin [6], but Miermont and Schweinsberg [38] have recently shown that a construction of this form is possible only for α = j .”
Exchangeable Fragmentation-Coalescence Processes and their Equilibrium Measures (2004, unknown): “Final version accepted on September”