Librería Portfolio

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

The theory of large deviations deals with rates at which probabilities of certain events decay as a natural parameter in the problem varies. This book, which is based on a graduate course on large deviations at the Courant Institute, focuses on three concrete sets of examples: (i) diffusions with small noise and the exit problem, (ii) large time behavior of Markov processes and their connection to the Feynman-Kac formula and the related large deviation behavior of the number of distinct sites visited by a random walk, and (iii) interacting particle systems, their scaling limits, and large deviations from their expected limits. For the most part the examples are worked out in detail, and in the process the subject of large deviations is developed.

The book will give the reader a flavor of how large deviation theory can help in problems that are not posed directly in terms of large deviations. The reader is assumed to have some familiarity with probability, Markov processes, and interacting particle systems.



Table of Contents

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Cover 1
Title page 4
Contents 6
Preface 8
Chapter 1. Introduction 10
1.1. Outline 10
1.2. Supplementary Material 11
Chapter 2. Basic Formulation 14
2.1. What Are Large Deviations? 14
2.2. Evaluation of Integrals 17
2.3. Contraction Principle 18
2.4. Simple Examples and Remarks 19
Chapter 3. Small Noise 22
3.1. The Exit Problem 22
3.2. Large Deviations of {??_{??,??}} 23
3.3. The Exit Problem 26
3.4. Superexponential Estimates 29
3.5. General Diffusion Processes 31
3.6. Short-Time Behavior of Diffusions 34
3.7. Supplementary Material 36
Chapter 4. Large Time 38
4.1. Introduction 38
4.2. Large Deviations and the Principal Eigenvalues 41
4.3. More General State Spaces 42
4.4. Dirichlet Eigenvalues 43
4.5. Lower Bound 44
4.6. Upper Bounds 46
4.7. The Role of Topology 49
4.8. Finishing Up 52
4.9. Remarks 52
Chapter 5. Hydrodynamic Scaling 54
5.1. From Classical Mechanics to Euler Equations 54
5.2. Simple Exclusion Processes 57
5.3. Symmetric Simple Exclusion 61
5.4. Weak Asymmetry 64
5.5. Large Deviations 71
Chapter 6. Self-Diffusion 80
6.1. Motion of a Tagged Particle 80
Chapter 7. Nongradient Systems 88
7.1. Multicolor Systems 88
7.2. Tightness Estimates 93
7.3. Approximations 97
7.4. Calculating Variances 100
7.5. Proofs 104
Chapter 8. Some Comments About TASEP 108
Bibliography 112
Back Cover 114
Table of Contents
Preface
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