The main purpose of the book is to introduce some progress on probability theory and its applications to physics, made by Chinese probabilists,especially by a group at Beijing Normal University in the past 15 years. Up tnull
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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. ProofofUniqueness 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. OptimalMarkovian 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 …… Part III. Equilibrium Particle Systems Part Ⅳ. Non-equilibrium Particle Systems