Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features:
. A careful examination of how a dynamical system can serve as a generator of stochastic processes
. Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes
. Several examples of analysis of extremes in a physical and geophysical context
. A final summary of the main results presented along with a guide to future research projects
. An appendix with software in Matlab® programming language to help readers to develop further understanding of the presented concepts
Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science.
VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK.
DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l'environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France.
ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal.
JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal.
MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK.
TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK.
MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA.
MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland.
SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.Autorentext
VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK.
DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l'environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France.
ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal.
JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal.
MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK.
TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK.
MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA.
MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland.
SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.
Inhalt
1 Introduction 1
1.1 A Transdisciplinary Research Area 1
1.2 Some Mathematical Ideas 4
1.3 Some Difficulties and Challenges in Studying Extremes 6
1.3.1 Finiteness of Data 6
1.3.2 Correlation and Clustering 8
1.3.3 Time Modulations and Noise 9
1.4 Extremes Observables and Dynamics 10
1.5 This Book 12
Acknowledgments 14
2 A Framework for Rare Events in Stochastic Processes and Dynamical Systems 17
2.1 Introducing Rare Events 17
2.2 Extremal Order Statistics 19
2.3 Extremes and Dynamics 20
3 Classical Extreme Value Theory 23
3.1 The i.i.d. Setting and the Classical Results 24
3.1.1 Block Maxima and the Generalized Extreme Value Distribution 24
3.1.2 Examples 26
3.1.3 Peaks Over Threshold and the Generalized Pareto Distribution 28
3.2 Stationary Sequences and Dependence Conditions 29
3.2.1 The Blocking Argument 30
3.2.2 The Appearance of Clusters of Exceedances 31
3.3 Convergence of Point Processes of Rare Events 32
3.3.1 Definitions and Notation 33
3.3.2 Absence of Clusters 35
3.3.3 Presence of Clusters 35
3.4 Elements of Declustering 37
4 Emergence of Extreme Value Laws for Dynamical Systems 39
4.1 Extremes for General Stationary Processes-an Upgrade Motivated by Dynamics 40
4.1.1 Notation 41
4.1.2 The New Conditions 42
4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 44
4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 46
4.2 Extreme Values for Dynamically Defined Stochastic Processes 51
4.2.1 Observables and Corresponding Extreme Value Laws 53
4.2.2 Extreme Value Laws for Uniformly Expanding Systems 57
4.2.3 Example Revisited 59
4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 61
4.3 Point Processes of Rare Events 62
4.3.1 Absence of Clustering 62
4.3.2 Presence of Clustering 63
4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 65
4.4 Conditions q(un), D3(un) Dp(un)* and Decay of Correlations 66
4.5 Specific Dynamical Systems Where the Dichotomy Applies 70
4.5.1 Rychlik Systems 70
4.5.2 Piecewise Expanding Maps in Higher Dimensions 71
4.6 Extreme Value Laws for Physical Observables 72
5 Hitting and Return Time Statistics 75
5.1 Introduction to Hitting and Return Time Statistics 75
5.1.1 Definition of Hitting and Return Time Statistics 76
5.2 HTS Versus RTS and Possible Limit Laws 77
5.3 The Link Between Hitting Times and Extreme Values 78
5.4 Uniformly Hyperbolic Systems 84
5.4.1 Gibbs Measures 85
5.4.2 First HTS Theorem 86
5.4.3 Markov Partitions 86
5.4.4 Two-Sided Shifts 88
5.4.5 Hyperbolic Diffeomorphisms 89
5.4.6 Additional Uniformly Hyperbolic Examples 90
5.5 Nonuniformly Hyperbolic Systems 91
5.5.1 Induced System 91
5.5.2 Intermittent Maps 92
5.5.3 Interval Maps with Critical Points 93
5.5.4 Higher Dimensional Examples of Nonuniform Hyperbolic Systems 94
5.6 Nonexponential Laws 95
6 Extreme Value Theory for Selected Dynamical Systems 97
6.1 Rare Events and Dynamical Systems 97
6.2 Introduction and Background on Extremes in Dynamical Systems 98
6.3 The Blocking Argument for Nonuniformly Expanding Systems 99
6.3.1 Assumptions on …