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
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