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E-raamat: Gaussian Processes on Trees: From Spin Glasses to Branching Brownian Motion

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(Rheinische Friedrich-Wilhelms-Universität Bonn)
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Gaussian Processes on Trees: From Spin Glasses to Branching Brownian MotionSuurem pilt
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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.

Arvustused

'The text is a very well-written presentation of the motivations and recent developments in the study of the extreme process of the BBM. This provides a perfect guide for any researcher interested in this field, especially those who are looking for a relatively quick introduction.' Bastien Mallein, Mathematical Reviews 'When discussing most of the questions, the author pays good attention to both ideas and techniques. He presents a large number of results, many of them are non-trivial limit theorems. Some results are classical in the eld, others are quite new, published very recently. While some of the results belong to the author, credit is given to several other contributors in the area. Besides the many results given with their proofs, the author includes useful bibliographical notes in the end of each chapter. The book ends with a comprehensive list of 117 references and Index. This is a well-written book on hot topics from modern stochastics and its applications. The book can be recommended to researchers and university graduate students.' Jordan M. Stoyanov, Zentralblatt MATH

Muu info

This book presents recent advances in branching Brownian motion from the perspective of extreme value theory and statistical physics, for graduates.
Preface vii
Acknowledgements x
1 Extreme Value Theory for iid Sequences
1(14)
1.1 Basic Issues
1(1)
1.2 Extremal Distributions
2(10)
1.3 Level-Crossings and kth Maxima
12(1)
1.4 Bibliographic Notes
13(2)
2 Extremal Processes
15(19)
2.1 Point Processes
15(3)
2.2 Laplace functionals
18(1)
2.3 Poisson Point Processes
19(2)
2.4 Convergence of Point Processes
21(8)
2.5 Point Processes of Extremes
29(4)
2.6 Bibliographic Notes
33(1)
3 Normal Sequences
34(11)
3.1 Normal Comparison
35(7)
3.2 Applications to Extremes
42(2)
3.3 Bibliographic Notes
44(1)
4 Spin Glasses
45(15)
4.1 Setting and Examples
45(2)
4.2 The REM
47(2)
4.3 The GREM, Two Levels
49(5)
4.4 Connection to Branching Brownian Motion
54(1)
4.5 The Galton--Watson Process
55(2)
4.6 The REM on the Galton--Watson Tree
57(2)
4.7 Bibliographic Notes
59(1)
5 Branching Brownian Motion
60(16)
5.1 Definition and Basics
60(1)
5.2 Rough Heuristics
61(2)
5.3 Recursion Relations
63(2)
5.4 The F-KPP Equation
65(2)
5.5 The Travelling Wave
67(3)
5.6 The Derivative Martingale
70(5)
5.7 Bibliographic Notes
75(1)
6 Bramson's Analysis of the F-KPP Equation
76(46)
6.1 Feynman--Kac Representation
76(4)
6.2 The Maximum Principle and its Applications
80(15)
6.3 Estimates on the Linear F-KPP Equation
95(3)
6.4 Brownian Bridges
98(4)
6.5 Hitting Probabilities of Curves
102(3)
6.6 Asymptotics of Solutions of the F-KPP Equation
105(7)
6.7 Convergence Results
112(9)
6.8 Bibliographic Notes
121(1)
7 The Extremal Process of BBM
122(23)
7.1 Limit Theorems for Solutions
122(5)
7.2 Existence of a Limiting Process
127(5)
7.3 Interpretation as Cluster Point Process
132(12)
7.4 Bibliographic Notes
144(1)
8 Full Extremal Process
145(8)
8.1 The Embedding
145(2)
8.2 Properties of the Embedding
147(2)
8.3 The q-Thinning
149(3)
8.4 Bibliographic Notes
152(1)
9 Variable Speed BBM
153(38)
9.1 The Construction
153(1)
9.2 Two-Speed BBM
154(22)
9.3 Universality Below the Straight Line
176(13)
9.4 Bibliographic Notes
189(2)
References 191(8)
Index 199
Anton Bovier is Professor of Mathematics at the University of Bonn. He is the author of more than 130 scientific papers and two monographs, Statistical Mechanics of Disordered Systems: A Mathematical Perspective (Cambridge, 2006) and Metastability: A Potential-Theoretic Approach (with Frank den Hollander, 2016). Bovier is a Fellow of the Institute of Mathematical Statistics and a member of the Clusters of Excellence, The Hausdorff Center for Mathematics and ImmunoSensation, both at the University of Bonn.
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