特征值，不等式和遍歷理論

前言

　　First， let us explain the precise meaning of the compressed title. The word "eigenvalues" means the first nontrivial Neumann or Dirichlet eigenvalues， or the principal eigenvalues. The word "inequalities" means the Poincare inequalities， the logarithmic Sobolev inequalities， the Nash inequalities， and so on.　Actually， the first eigenvalues can be described by some Poincare inequalities， and so the second topic has a wider range than the first one. Next， for a Markov process， corresponding to its operator， each inequality describes a type of ergodicity. Thus， study of the inequalities and their relations provides a way to develop the ergodic theory for Markov processes. Due to these facts， from a probabilistic point of view， the book can also be regarded as a study of "ergodic convergence rates of Markov processes，" which could serve as an alternative title of the book. However， this book is aimed at a larger class of readers， not only probabilists.　　The importance of these topics should be obvious. On the one hand， the first eigenvalue is the leading term in the spectrum， which plays an important role in almost every branch of mathematics. On the other hand， the ergodic convergence rates constitute a recent research area in the theory of Markov processes. This study has a very wide range of applications. In particular， it provides a tool to describe the phase transitions and the effectiveness of random algorithms， which are now a very fashionable research area.　　……

內(nèi)容概要

　　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.

書籍目錄

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

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