Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics.
Author(s): Anton Bovier
Series: Cambridge Studies in Advanced Mathematics 163
Publisher: Cambridge University Press
Year: 2016
Language: English
Pages: 211
Contents......Page 6
Preface......Page 7
Acknowledgements......Page 11
1.1 Basic Issues......Page 12
1.2 Extremal Distributions......Page 13
1.3 Level-Crossings and kth Maxima......Page 23
1.4 Bibliographic Notes......Page 24
2.1 Point Processes......Page 26
2.2 Laplace functionals......Page 29
2.3 Poisson Point Processes......Page 30
2.4 Convergence of Point Processes......Page 32
2.5 Point Processes of Extremes......Page 40
2.6 Bibliographic Notes......Page 44
3 Normal Sequences......Page 45
3.1 Normal Comparison......Page 46
3.2 Applications to Extremes......Page 53
3.3 Bibliographic Notes......Page 55
4.1 Setting and Examples......Page 56
4.2 The REM......Page 58
4.3 The GREM, Two Levels......Page 60
4.4 Connection to Branching Brownian Motion......Page 65
4.5 The Galton–Watson Process......Page 66
4.6 The REM on the Galton–Watson Tree......Page 68
4.7 Bibliographic Notes......Page 70
5.1 Definition and Basics......Page 71
5.2 Rough Heuristics......Page 72
5.3 Recursion Relations......Page 74
5.4 The F-KPP Equation......Page 76
5.5 The Travelling Wave......Page 78
5.6 The Derivative Martingale......Page 81
5.7 Bibliographic Notes......Page 86
6.1 Feynman–Kac Representation......Page 87
6.2 The Maximum Principle and its Applications......Page 91
6.3 Estimates on the Linear F-KPP Equation......Page 106
6.4 Brownian Bridges......Page 109
6.5 Hitting Probabilities of Curves......Page 113
6.6 Asymptotics of Solutions of the F-KPP Equation......Page 116
6.7 Convergence Results......Page 123
6.8 Bibliographic Notes......Page 132
7.1 Limit Theorems for Solutions......Page 133
7.2 Existence of a Limiting Process......Page 138
7.3 Interpretation as Cluster Point Process......Page 143
7.4 Bibliographic Notes......Page 155
8.1 The Embedding......Page 156
8.2 Properties of the Embedding......Page 158
8.3 The q-Thinning......Page 160
8.4 Bibliographic Notes......Page 163
9.1 The Construction......Page 164
9.2 Two-Speed BBM......Page 165
9.3 Universality Below the Straight Line......Page 187
9.4 Bibliographic Notes......Page 200
References......Page 202
Index......Page 210
