陈木法编著的《特征值、不等式和遍历理论》详细讲述了应用广泛众多读者感兴趣的问题———矩阵的谱隙和微分算子，提供了描述相变换和随机算术效应的工具。每章以综述开始，为了吸引更多的非专业人士，通过简单例子引入观点而不是技巧证明。后面的一些章节很自然地将读者引入这个领域的问题和应用中，如经济的随机模型等。

读者对象：数学、统计、经济领域的学生和相关的科研人员。

Preface

Acknowledgments

Chapter 1 An Overview of the Book

 1.1 Introduction

 1.2 New variational formula for the first eigenvalue

 1.3 Basic inequalities and new forms of Cheeger's constants

 1.4 A new picture of ergodic theory and explicit criteria

Chapter 2 Optimal Markovian Couplings

 2.1 Couplings and Markovian couplings

 2.2 Optimality with respect to distances

 2.3 Optimality with respect to closed functions

 2.4 Applications of coupling methods

Chapter 3 New Variational Formulas for the First Eigenvalue

 3.1 Background

 3.2 Partial proof in the discrete case

 3.3 The three steps of the proof in the geometric case

 3,4 Two difficulties

 3.5 The final step of the proof of the formula

 3.6 Comments on different methods

 3.7 Proof in the discrete case (continued)

 3.8 The first Dirichlet eigenvalue

Chapter 4 Generalized Cheeger's Method

 4.1 Cheeger's method

 4.2 A generalization

 4.3 New results

 4.4 Splitting technique and existence criterion

 4.5 Proof of Theorem 4.4

 4.6 Logarithmic Sobolev inequality

 4.7 Upper bounds

 4.8 Nash inequality

 4.9 Birth-death processes

Chapter 5 Ten Explicit Criteria in Dimension One

 5.1 Three traditional types of ergodicity

 5.2 The first (nontrivial) eigenvalue (spectral gap)

 5.3 The first eigenvalues and exponentially ergodic rate

 5.4 Explicit criteria

 5.5 Exponential ergodicity for single birth processes

 5.6 Strong ergodicity

Chapter 6 Poincar-Type Inequalities in Dimension One

 6.1 Introduction

 6.2 Ordinary Poincar inequalities

 6.3 Extension: normed linear spaces

 6.4 Neumann case: Orlicz spaces

 6.5 Nash inequality and Sobolev-type inequality

 6.6 Logarithmic Sobolev inequality

 6.7 Partial proofs of Theorem 6.1

Chapter 7 Functional Inequalities

 7.1 Statement of results

 7.2 Sketch of the proofs

 7.3 Comparison with Cheeger's method

 7.4 General convergence speed

 7.5 Two functional inequalities

 7.6 Algebraic convergence

 7.7 General (irreversible) case

Chapter 8 A Diagram of Nine Types of Ergodicity

 8.1 Statements of results

 8.2 Applications and comments

 8.3 Proof of Theorem 1.9

Chapter 9 Reactlon-Diffusion Processes

 9.1 The models

 9.2 Finite-dimensional case

 9.3 Construction of the processes

 9.4 Ergodicity and phase transitions

 9.5 Hydrodynamic limits

Chapter 10 Stochastic Models of Economic Optimization

 10.1 Input-output method

 10.2 L.K. Hua's fundamental theorem

 10.3 Stochastic model without consumption

 10.4 Stochastic model with consumption

 10.5 Proof of Theorem 10.4

Appendix A Some Elementary Lemmas

Appendix B Examples of the Ising Model on Two to Four Sites

 B.1 The model

 B.2 Distance based on symmetry: two sites

 B.3 Reduction: three sites

 B.4 Modification: four sites

References

Author Index

Subject Index