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48
Homogeneous fragmentation processes
, 2000
"... The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by som ..."
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Cited by 29 (4 self)
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The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by some exchangeable measure on the set of partitions of N, and results from the combination of two different phenomena: a continuous erosion and sudden dislocations. In particular, we determine the class of fragmentation measures which can arise in this setting, and investigate the evolution of the size of the fragment that contains a point pick at random at the initial time.
Superprocesses with dependent spatial motion and general branching densities
 Electronic Journal of Probability
, 2001
"... We construct a class of superprocesses by taking the high density limit of a sequence of interactingbranching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching density is given by an arbitrary bounded nonnegative Borel funct ..."
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Cited by 28 (16 self)
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We construct a class of superprocesses by taking the high density limit of a sequence of interactingbranching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching density is given by an arbitrary bounded nonnegative Borel function, and the superprocess is characterized by a martingale problem as a diffusion process with state space M(IR), improving and extending considerably the construction of Wang (1997, 1998). It is then proved in a special case that a suitable rescaled process of the superprocess converges to the usual super Brownian motion. An extension to measurevalued branching catalysts is also discussed. AMS Subject Classifications: Primary 60J80, 60G57; Secondary 60J35 Key words and phrases: superprocess, interactingbranching particle system, diffusion process, martingale problem, dual process, rescaled limit, measurevalued catalyst.
Inhomogeneous Continuum Random Trees and the Entrance Boundary of the Additive Coalescent
 PROBAB. TH. REL. FIELDS
, 1998
"... Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; ..."
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Cited by 27 (12 self)
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Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i + x j at rate x i + x j . Aldous and Pitman (1998) showed that a version of this process starting from time \Gamma1 with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees.
Clustering in coagulationfragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and ZeroOne law
"... The equilibrium distribution of a reversible coagulationfragmentation process (CFP) and the joint distribution of components of a random combinatorial structure are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (=compon ..."
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Cited by 19 (11 self)
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The equilibrium distribution of a reversible coagulationfragmentation process (CFP) and the joint distribution of components of a random combinatorial structure are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (=components) in the case a(k) = kp−1, k ≥ 1, p> 0, where a(k), k ≥ 1 is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic random combinatorial structures (RCS’s) and for RCS’s, corresponding to the case p < 0. 1 Summary. Our main result is a central limit theorem (Theorem 2.4) for the number of groups at steady state for a class of reversible CFP’s and for the corresponding class of RCS’s. In Section 2, we provide a definition of a reversible kCFP admitting interactions of up to k groups, as a generalization of the standard 2CFP. The steady state of the processes considered is fully defined by a parameter function a ≥ 0 on the set of integers. It was observed by Kelly ([11], p. 183) that for all 2 ≤ k ≤ N the kCFP’s have the same invariant measure on the set of partitions of a given
Exchangeable partitions derived from Markovian coalescents
 Adv. Appl. Probab
, 2006
"... Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. Möhle described the recursion which determines the generalization of the Ewens sampling formula when the ..."
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Cited by 19 (3 self)
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Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. Möhle described the recursion which determines the generalization of the Ewens sampling formula when the lines of descent are governed by a coalescent with multiple collisions. In [7] authors exploit an analogy with the theory of regenerative composition and partition structures, and provide various characterizations of the associated exchangeable random partitions. This paper gives parallel results for the further generalized model with lines of descent following a coalescent with simultaneous multiple collisions. 1
Two coalescents derived from the ranges of stable subordinators
 Electron. J. Probab
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The Entrance Boundary of the Multiplicative Coalescent
 ELECTRON. J. PROBAB
, 1998
"... The multiplicative coalescent X(t) is a l²valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a standard version of this process ..."
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Cited by 15 (7 self)
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The multiplicative coalescent X(t) is a l²valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a standard version of this process starting with infinitesimally small clusters at time \Gamma1. In this paper, stochastic calculus techniques are used to describe all versions (X(t); \Gamma1 ! t ! 1) of the multiplicative coalescent. Roughly, an extreme version is specified by translation and scale parameters, and a vector c 2 l 3 of relative sizes of large clusters at time \Gamma1. Such a version may be characterized in three ways: via its t ! \Gamma1 behavior, via a representation of the marginal distribution X(t) in terms of excursionlengths of a L'evytype process, or via a weak limit of processes derived from the standard multiplicative coalescent using a "coloring" construction.
Random mappings, forests, and subsets associated with AbelCayleyHurwitz multinomial expansions
, 2001
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A Vervaatlike path transformation for the reflected Brownian bridge conditioned on its local time at 0
, 1999
"... We describe a Vervaatlike path transformation for the reflected Brownian bridge conditioned on its local time at 0: up to random shifts, this process equals the two processes constructed from a Brownian bridge and a Brownian excursion by adding a drift and then taking the excursions over the cur ..."
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Cited by 13 (2 self)
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We describe a Vervaatlike path transformation for the reflected Brownian bridge conditioned on its local time at 0: up to random shifts, this process equals the two processes constructed from a Brownian bridge and a Brownian excursion by adding a drift and then taking the excursions over the current minimum. As a consequence, these three processes have the same occupation measure, which is easily found. The three processes arise as limits, in three different ways, of profiles associated to hashing with linear probing, or, equivalently, to parking functions.