特徵值，不等式和遍歷理論(英文)（簡體書）

This book surveys， in a popular way， the main progress made in the field by our group. It consists of ten chapters plus two appendixes. The first chapter is an overview of the second to the eighth ones. Mainly， we study several different inequalities or different types of convergence by using three mathematical tools: a probabilistic tool， the coupling methods （Chapters 2 and 3）; a generalized Cheeger''s method originating in Riemannian geometry （Chapter 4）; and an approach coming from potential theory and harmonic analysis （Chapters 6 and 7）. The explicit criteria for different types of convergence and the explicit estimates of the convergence rates （or the optimal constants in the inequalities） in dimension one are given in Chapters 5 and 6; some generalizations are given in Chapter 7. The proofs of a diagram of nine types of ergodicity （Theorem 1.9） are presented in Chapter 8.．本書詳細講述了應用廣泛眾多讀者感興趣的問題―――矩陣的譜隙和微分算子，提供了描述相變換和隨機算術效應的工具。每章以綜述開始，為了吸引更多的非專業人士，通過簡單例子引入觀點而不是技巧證明。後面的一些章節很自然地將讀者引入這個領域的問題和應用中，如經濟的隨機模型等。讀者對象：數學、統計、經濟領域的學生和相關的科研人員。

《特征值,不等式和遍歷理論》讀者對象為數學、統計、經濟領域的學生和相關的科研人員。

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 Poincare-Type Inequalities in Dimension One
6.1 Introduction
6.2 Ordinary Poincare 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 Reaction-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
References
Author Index
Subject Index．