Large Deviations ( Courant Lecture Notes )

Publication series ：Courant Lecture Notes

Author： S. R. S. Varadhan  

Publisher： American Mathematical Society‎

Publication year： 2016

E-ISBN: 9781470435851

P-ISBN(Paperback): 9780821840863

Subject： O211.62 Markov process

Keyword： 暂无分类

Language： ENG

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Large Deviations

Description

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.

Chapter

Cover

Title page

Contents

Preface

Chapter 1. Introduction

1.1. Outline

1.2. Supplementary Material

Chapter 2. Basic Formulation

2.1. What Are Large Deviations?

2.2. Evaluation of Integrals

2.3. Contraction Principle

2.4. Simple Examples and Remarks

Chapter 3. Small Noise

3.1. The Exit Problem

3.2. Large Deviations of {𝑃_{𝜀,𝑥}}

3.3. The Exit Problem

3.4. Superexponential Estimates

3.5. General Diffusion Processes

3.6. Short-Time Behavior of Diffusions

3.7. Supplementary Material

Chapter 4. Large Time

4.1. Introduction

4.2. Large Deviations and the Principal Eigenvalues

4.3. More General State Spaces

4.4. Dirichlet Eigenvalues

4.5. Lower Bound

4.6. Upper Bounds

4.7. The Role of Topology

4.8. Finishing Up

4.9. Remarks

Chapter 5. Hydrodynamic Scaling

5.1. From Classical Mechanics to Euler Equations

5.2. Simple Exclusion Processes

5.3. Symmetric Simple Exclusion

5.4. Weak Asymmetry

5.5. Large Deviations

Chapter 6. Self-Diffusion

6.1. Motion of a Tagged Particle

Chapter 7. Nongradient Systems

7.1. Multicolor Systems

7.2. Tightness Estimates

7.3. Approximations

7.4. Calculating Variances

7.5. Proofs

Chapter 8. Some Comments About TASEP

Bibliography

Back Cover

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