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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 46 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Gelation in coagulation and fragmentation models
- Comm. Math. Phys
"... Abstract. Rates of decay for the total mass of the solutions to Smoluchovski’s equation with homogeneous kernels of degree λ> 1 are proved. That implies that gelation always occurs. Morrey estimates from below and from above on solutions around the gelation time are also obtained which are in agr ..."
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Cited by 45 (8 self)
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Abstract. Rates of decay for the total mass of the solutions to Smoluchovski’s equation with homogeneous kernels of degree λ> 1 are proved. That implies that gelation always occurs. Morrey estimates from below and from above on solutions around the gelation time are also obtained which are in agreement with previously known formal results on the profile of solutions at gelling time. The same techniques are applied to the coagulation-fragmentation model for which gelation is established in some particular cases. Key words: Smoluchowski’s coagulation equation, decay rate, gelation, profile at gelling time, Morrey-Campanato norms, existence of solutions. 1 Introduction and Main results The purpose of this work is to investigate gelling transition in coagulation and fragmentation model. The simplest model is the Smoluchowski coagulation equation describing irreversible aggregation processes between particles which coalesce and form larger and larger clusters.
Self-similar fragmentations
, 2000
"... We introduce a probabilistic model that is meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such a process is determined by its index of self-similarity α ∈ R, a rate of erosion c ≥ 0, and a so-called Lév ..."
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Cited by 37 (9 self)
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We introduce a probabilistic model that is meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such a process is determined by its index of self-similarity α ∈ R, a rate of erosion c ≥ 0, and a so-called Lévy measure that accounts for sudden dislocations. The key of the analysis is provided by a transformation of self-similar fragmentations which enables us to reduce the study to the homogeneous case α = 0 which is treated in [6].
Exponential decay towards equilibrium for the inhomogeneous Aizenman-Bak model
- Comm. Math. Phys
"... In this work, we show how the entropy method enables to get in an elementary way (and without linearization) estimates of exponential decay towards equilibrium for solutions of reaction-diusion equations corresponding to a reversible reaction. Explicit rates of convergence combining the dissipative ..."
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Cited by 36 (18 self)
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In this work, we show how the entropy method enables to get in an elementary way (and without linearization) estimates of exponential decay towards equilibrium for solutions of reaction-diusion equations corresponding to a reversible reaction. Explicit rates of convergence combining the dissipative eects of diusion and reaction are given.
Phase transition for parking blocks, Brownian excursion and coalescence
, 2005
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Small-time behavior of beta coalescents
, 2008
"... For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λ-coalescent in which Λ is the Beta(2−α, α) dist ..."
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Cited by 32 (12 self)
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For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λ-coalescent in which Λ is the Beta(2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λ-coalescents for which Λ has the same asymptotic behavior near zero as the Beta(2 − α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of Λ-coalescents.
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.
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 \Delta-valued 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 \Delta-valued 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.
Asymptotic laws for nonconservative self-similar fragmentations
, 2008
"... We consider a self-similar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probability. W ..."
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Cited by 27 (4 self)
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We consider a self-similar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probability. We show that under certain conditions the typical size in the ensemble is of the order t −1/α and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law.
Dust and self-similarity for the Smoluchowski coagulation equation
, 2004
"... We establish the well-posedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and lo ..."
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Cited by 26 (4 self)
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We establish the well-posedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and loses mass in finite time (gelation phenomena) when λ> 1. We then extend the existence result to a measure framework allowing dust source term. In that case we prove that the income dust instantaneously aggregates and the solution does not contain dust phase. On the other hand, we investigate the qualitative properties of self-similar solutions to the Smoluchowski’s coagulation equation when λ < 1. We prove regularity results and sharp uniform small and large size behavior for the self-similar profiles.
