Chapter 1 An Overview of the Book 1
1.1 Introduction 1
1.2 New variational formula for the first eigenvalue 3
1.3 Basic inequalities and new forms of Cheeger's constants 10
1.4 A new picture of ergodic theory and explicit criteria 12
Chapter 2 Optimal Markovian Couplings 17
2.1 Couplings and Markovian couplings 17
2.2 Optimality with respect to distances 26
2.3 Optimality with respect to closed functions 31
2.4 Applications of coupling methods 33
Chapter 3 New Variational Formulas for the First Eigenvalue 41
3.1 Background 41
3.2 Partial proof in the discrete case 43
3.3 The three steps of the proof in the geometric case 47
3.4 Two difficulties 50
3.5 The final step of the proof of the formula 54
3.6 Comments on different methods 56
3.7 Proof in the discrete case（continued） 58
3.8 The first Dirichlet eigenvalue 62
Chapter 4 Generalized Cheeger's Method 67
4.1 Cheeger's method 67
4.2 A generalization 68
4.3 New results 70
4.4 Splitting technique and existence criterion 71
4.5 Proof of Theorem 4.4 77
4.6 Logarithmic Sobolev inequality 80
4.7 Upper bounds 83
4.8 Nash inequality 85
4.9 Birth-death processes 87
Chapter 5 Ten Explicit Criteria in Dimension One 89
5.1 Three traditional types of ergodicity 89
5.2 The first（nontrivial）eigenvalue（spectral gap） 92
5.3 The first eigenvalues and exponentially ergodic rate 95
5.4 Explicit criteria 98
5.5 Exponential ergodicity for single birth processes 100
5.6 Strong ergodicity 106
Chapter 6 Poincaré-Type Inequalities in Dimension One 113
6.1 Introduction 113
6.2 Ordinary Poincaré inequalities 115
6.3 Extension:normed linear spaces 119
6.4 Neumann case:Orlicz spaces 121
6.5 Nash inequality and Sobolev-type inequality 123
6.6 Logarithmic Sobolev inequality 125
6.7 Partial proofs of Theorem 6.1 127
Chapter 7 Functional Inequalities 131
7.1 Statement of results 131
7.2 Sketch of the proofs 137
7.3 Comparison with Cheeger's method 140
7.4 General convergenee speed 142
7.5 Two functional inequalities 143
7.6 Algebraic convergenee 145
7.7 General（irreversible）case 147
Chapter 8 A Diagram of Nine Types of Ergodicity 149
8.1 Statements of results 149
8.2 Applications and comments 152
8.3 Proof of Theorem 1.9 155
Chapter 9 Reaction-Diffusion Processes 163
9.1 The models 164
9.2 Finite-dimensional case 166
9.3 Construction of the processes 170
9.4 Ergodicity and phase transitions 175
9.5 Hydrodynamic limits 177
Chapter 10 Stochastic Models of Economic Optimization 181
10.1 Input-output method 181
10.2 L.K.Hua's fundamental theorem 182
10.3 Stochastic model without consumption 186
10.4 Stochastic model with consumption 188
10.5 Proof of Theorem 10.4 190
Appendix A Some Elementary Lemmas 193
Appendix B Examples of the Ising Model on Two to Four Sites 197
B.1 The model 197
B.2 Distance based on symmetry:two sites 198
B.3 Reduction:three sites 200
B.4 Modification:four sites 205
References 209
Author Index 223
Subject Index 226