Random Fragmentation and Coagulation Processes

Jean Bertoin0521867282, 9780521867283, 9780511247163

Fragmentation and coagulation are two natural phenomena that can be observed in many sciences and at a great variety of scales-from, for example, DNA fragmentation to formation of planets by accretion. This book, by the author of the acclaimed LГcvy Processes, is the first comprehensive theoretical account of mathematical models for situations where either phenomenon occurs randomly and repeatedly as time passes. This self-contained treatment develops the models in a way that makes recent developments in the field accessible. Each chapter ends with a comments section in which important aspects not discussed in the main part of the text (often because the discussion would have been too technical and/or lengthy) are addressed and precise references are given. Written for readers with a solid background in probability, its careful exposition allows graduate students, as well as working mathematicians, to approach the material with confidence.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Series-title……Page 4
Title……Page 5
Copyright……Page 6
Contents……Page 7
Introduction……Page 11
1.1 Construction of fragmentation chains……Page 16
1.1.1 Preliminaries on Markov chains……Page 17
1.1.2 Branching Markov chains……Page 21
1.1.3 Fragmentation chains……Page 26
1.2 Genealogical structure……Page 33
1.2.1 The tree of generations……Page 34
1.2.2 Malthusian hypotheses and the intrinsic martingale……Page 36
1.2.3 A randomly tagged branch……Page 41
1.3 Extinction and formation of dust for Alpha < 0……Page 47
1.3.1 Extinction……Page 48
1.3.2 Formation of dust……Page 50
1.4 Some strong laws for………Page 53
1.4.1 A variation of the law of large numbers……Page 54
1.4.2 The homogeneous case (Alpha = 0)……Page 56
1.4.3 The case Alpha > 0……Page 59
1.4.4 Another strong law via renewal theory……Page 63
1.5 Additive martingales (homogeneous case Alpha = 0)……Page 65
1.5.1 Convergence of additive martingales……Page 66
1.5.2 Some applications……Page 68
1.6 Comments……Page 72
2.1.1 Partitions of a unit mass……Page 76
2.1.2 Interval-partitions……Page 78
2.1.3 Size-biased sampling and reordering……Page 81
2.2 Random mass-partitions and Poisson measures……Page 84
2.2.1 Multidimensional Dirichlet distributions……Page 85
2.2.2 Some preliminaries on Poisson random measures……Page 88
2.2.3 Mass-partitions induced by Poisson measures……Page 91
2.2.4 Gamma subordinators and Dirichlet processes……Page 97
2.2.5 Stable subordinators and Poisson-Dirichlet partitions……Page 100
2.3 Exchangeable random partitions……Page 104
2.3.1 Some definition……Page 105
2.3.2 Kingman’s theory……Page 107
2.3.3 Exchangeable partition probability functions……Page 115
2.4 Comments……Page 120
3.1 Homogeneous fragmentation processes……Page 122
3.1.1 Fragmentation of partitions……Page 124
3.1.2 Homogeneous fragmentation as Markov processes……Page 129
3.1.3 Poissonian structure……Page 134
3.2 Asymptotic frequencies……Page 135
3.2.1 Erosion and dislocation……Page 136
3.2.2 Subordinator representation of the tagged fragment……Page 142
3.2.3 Lévy-Itô decomposition of the tagged fragment……Page 150
3.3 Self-similar fragmentations……Page 154
3.3.1 Definition and first properties……Page 155
3.3.2 Changing the index of self-similarity……Page 159
3.3.3 Mass-fragmentations……Page 162
3.4 Comments……Page 170
4.1.1 Genealogy of populations in the Wright-Fisher model……Page 173
4.1.2 Construction of Kingman’s coalescent……Page 175
4.1.3 Interval representation of Kingman’s coalescent……Page 181
4.2 Simultaneous and multiple coagulations……Page 183
4.2.1 Coagulation of partitions……Page 184
4.2.2 Exchangeable coalescents and coagulation rates……Page 187
4.2.3 Poissonian construction……Page 189
4.2.4 Characterization of coagulation rates……Page 191
4.3.1 Markov property……Page 195
4.3.2 Dust in exchangeable mass-coalescents……Page 197
4.4 Simple coalescents and flows of bridges……Page 199
4.4.1 Compositions of simple bridges……Page 200
4.4.2 Flows of bridges and coagulation……Page 206
4.4.3 The dual flow and a population model……Page 210
4.4.4 The Bolthausen-Sznitman coalescent……Page 215
4.5 Comments……Page 221
5.1 Stochastic coalescence……Page 224
5.1.1 Coalescent chains……Page 225
5.1.2 Extension to infinite systems……Page 228
5.2 Hydrodynamic behavior and Smoluchowski’s equations……Page 236
5.2.1 The multiplicative kernel……Page 237
5.2.2 Sub-multiplicative kernels……Page 245
5.3.1 Some basic properties……Page 254
5.3.2 Coagulation of trees in a random forest……Page 257
5.3.3 The standard additive coalescent……Page 264
5.3.4 A dual fragmentation process……Page 268
5.4 Comments……Page 271
References……Page 274
List of symbols……Page 286
Index……Page 289