Preface to the First Edition
Preface to the Second Edition
Chapter 0. An Overview of the Book: Starting From Markov Chains
0.1. Three Classical Problems for Markov Chains
0.2. Probability Metrics and Coupling Methods
0.3. Reversible Markov Chains
0.4. Large Deviations and Spectral Gap
0.5. Equilibrium Particle Systems
0.6. Non-equilibrium Particle Systems
Part I. General Jump Processes
Chapter 1. Transition Function and its Laplace Transform
1.1. Basic Properties of Transition Function
1.2. The q-Pair
1.3. Differentiability
1.4. Laplace Transforms
1.5. Appendix
1.6. Notes
Chapter 2. Existence and Simple Constructions of Jump Processes
2.1. Minimal Nonnegative Solutions
2.2. Kolmogorov Equations and Minimal Jump Process
2.3. Some Sufficient Conditions for Uniqueness
2.4. Kolmogorov Equations and q-Condition
2.5. Entrance Space and Exit Space
2.6. Construction of q-Processes with Single-Exit q-Pair
2.7. Notes
Chapter 3. Uniqueness Criteria
3.1. Uniqueness Criteria Based on Kolmogorov Equations
3.2. Uniqueness Criterion and Applications
3.3. Some Lemmas
3.4. Proof of Uniqueness Criterion
3.5. Notes
Chapter 4. Recurrence, Ergodicity and Invariant Measures
4.1. Weak Convergence
4.2. General Results
4.3. Markov Chains: Time-discrete Case
4.4. Markov Chains: Time-continuous Case
4.5. Single Birth Processes
4.6. Invariant Measures
4.7. Notes
Chapter 5. Probability Metrics and Coupling Methods
5.1. Minimum LP-Metric
5.2. Marginality and Regularity
5.3. Successful Coupling and Ergodicity
5.4. Optimal Markovian Couplings
5.5. Monotonicity
5.6. Examples
5.7. Notes
Part II. Symmetrizable Jump Processes
Chapter 6. Symmetrizable Jump Processes and Dirichlet Forms
6.1. Reversible Markov Processes
6.2. Existence
6.3. Equivalence of Backward and Forward Kolmogorov Equations
6.4. General Representation of Jump Processes
6.5. Existence of Honest Reversible Jump Processes
6.6. Uniqueness Criteria
6.7. Basic Dirichlet Form
6.8. Regularity, Extension and Uniqueness
6.9. Notes
Chapter 7. Field Theory
7.1. Field Theory
7.2. Lattice Field
7.3. Electric Field
7.4. Transience of Symmetrizable Markov Chains
7.5. Random Walk on Lattice Fractals
7.6. A Comparison Theorem
7.7. Notes
Chapter 8. Large Deviations
8.1. Introduction to Large Deviations
8.2. Rate Function
8.3. Upper Estimates
8.4. Notes
Chapter 9. Spectral Gap
9.1. General Case: an Equivalence
9.2. Coupling and Distance Method
9.3. Birth-Death Processes
9.4. Splitting Procedure and Existence Criterion
9.5. Cheeger's Approach and Isoperimetric Constants
9.6. Notes
Part III. Equilibrium Particle Systems
Chapter 10. Random Fields
10.1. Introduction
10.2. Existence
10.3. Uniqueness
10.4. Phase Transition: Peierls Method
10.5. Ising Model on Lattice Fractals
10.6. Reflection Positivity and Phase Transitions
10.7. Proof of the Chess-Board Estimates
10.8. Notes
Chapter 11. Reversible Spin Processes and Exclusion Processes
11.1. Potentiality for Some Speed Functions
11.2. Constructions of Gibbs States
11.3. Criteria for Reversibility
11.4. Notes
Chapter 12. Yang-Mills Lattice Field
12.1. Background
12.2. Spin Processes from Yang-Mills Lattice Fields
12.3. Diffusion Processes from Yang-Mills Lattice Fields
12.4. Notes
Part IV. Non-equilibrium Particle Systems
Chapter 13. Constructions of the Processes
13.1. Existence Theorems for the Processes
13.2. Existence Theorem for Reaction-Diffusion Processes
13.3. Uniqueness Theorems for the Processes
13.4. Examples
13.5. Appendix
13.6. Notes
Chapter 14. Existence of Stationary Distributions and Ergodicity
14.1. General Results
14.2. Ergodicity for Polynomial Model
14.3. Reversible Reaction-Diffusion Processes
14.4. Notes
Chapter 15. Phase Transitions
15.1. Duality
15.2. Linear Growth Model
15.3. Reaction-Diffusion Processes with Absorbing State
15.4. Mean Field Method
15.5. Notes
Chapter 16. Hydrodynamic Limits
16.1. Introduction: Main Results
16.2. Preliminaries
16.3. Proof of Theorem 16.1
16.4. Proof of Theorem 16.3
16.5. Notes
Bibliography
Author Index
Subject Index