| Introduction 1 Self-similar fragmentation chains 1.1 Construction of fragmentation chains 1.1.1 Preliminaries on Markov chains 1.1.2 Branching Markov chains 1.1.3 Fragmentation chains 1.2 Genealogical structure 1.2.1 The tree of generations 1.2.2 Malthusian hypotheses and the intrinsic martingale 1.2.3 A randomly tagged branch 1.3 Extinction and formation of dust for a＜0 1.3.1 Extinction 1.3.2 Formation of dust 1.4 Some strong laws for a＞0 1.4.1 A variation of the law of large numbers 1.4.2 The homogeneous case (a=0) 1.4.3 The case a＞0 1.4.4 Another strong law via renewal theory 1.5 Additive martingales (homogeneous case a=0) 1.5.1 Convergence of additive martingales 1.5.2 Some applications 1.6 Comments 2 Random partitions 2.1 Mass-partitions 2.1.1 Partitions of a unit mass 2.1.2 Interval-partitions 2.1.3 Size-biased sampling and reordering 2.2 Random mass-partitions and Poisson measures 2.2.1 Multidimensional Dirichlet distributions 2.2.2 Some preliminaries on Poisson random measures 2.2.3 Mass-partitions induced by Poisson measures 2.2.4 Gamma subordinators and Dirichlet processes 2.2.5 Stable subordinators and Poisson-Dirichlet partitions 2.3 Exchangeable random partitions 2.3.1 Some definition 2.3.2 Kingman's theory 2.3.3 Exchangeable partition probability functions 2.4 Comments 3 Exchangeable fragmentations 3.1 Homogeneous fragmentation processes 3.1.1 Fragmentation of partitions 3.1.2 Homogeneous fragmentation as Markov processes 3.1.3 Poissonian structure 3.2 Asymptotic frequencies 3.2.1 Erosion and dislocation 3.2.2 Subordinator representation of the tagged fragment 3.2.3 Levy-Ito decomposition of the tagged fragment 3.3 Self-similar fragmentations 3.3.1 Definition and first properties 3.3.2 Changing the index of self-similarity 3.3.3 Mass-fragmentations 3.4 Comments 4 Exchangeable coalescents 4.1 Kingman's coalescent 4.1.1 Genealogy of populations in the Wright-Fisher model 4.1.2 Construction of Kingman's coalescent 4.1.3 Interval representation of Kingman's coalescent 4.2 Simultaneous and multiple coagulations 4.2.1 Coagulation of partitions 4.2.2 Exchangeable coalescents and coagulation rates 4.2.3 Poissonian construction 4.2.4 Characterization of coagulation rates 4.3 Exchangeable mass-coalescents 4.3.1 Markov property 4.3.2 Dust in exchangeable mass-coalescents …… 5 Asymptotic regimes in stochastic coalescence References List of symbols Index |
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