This book is representative of the work of Chinese probabilists on probability theory and its applications in physics. It presents a unique treatment of general Markov jump processes: uniqueness, various types of ergodicity, Markovian couplings, reversibility, spectral gap, etc. It also deals with a typical class of non-equilibrium particle systems, including the typical Schlögl model taken from statistical physics. The constructions, ergodicity and phase transitions for this class of Markov interacting particle systems, namely, reaction–diffusion processes, are presented. In this new edition, a large part of the text has been updated and two-and-a-half chapters have been rewritten. The book is self-contained and can be used in a course on stochastic processes for graduate students.
Author(s): Mu-Fa Chen
Edition: 2
Publisher: World Scientific Publishing Company
Year: 2004
Language: English
Pages: 610
From Markov Chains to Non-equilibrium Particle Systems......Page 4
Contents......Page 6
Preface to the First Edition......Page 10
Preface to the Second Edition......Page 12
0.1. Three Classical Problems for Markov Chains......Page 14
0.2. Probability Metrics and Coupling Methods......Page 19
0.3. Reversible Markov Chains......Page 26
0.4. Large Deviations and Spectral Gap......Page 28
0.5. Equilibrium Particle Systems......Page 30
0.6. Non-equilibrium Particle Systems......Page 32
Part I. General Jump Processes......Page 34
1.1. Basic Properties of Transition Function......Page 36
1.2. The q-Pair......Page 40
1.3. Differentiability......Page 51
1.4. Laplace Transforms......Page 64
1.5. Appendix......Page 70
1.6. Notes......Page 74
2.1. Minimal Nonnegative Solutions......Page 75
2.2. Kolmogorov Equations and Minimal Jump Process......Page 83
2.3. Some Sufficient Conditions for Uniqueness......Page 92
2.4. Kolmogorov Equations and q-Condition......Page 98
2.5. Entrance Space and Exit Space......Page 101
2.6. Construction of q-Processes with Single-Exit q-Pair......Page 106
2.7. Notes......Page 109
3.1. Uniqueness Criteria Based on Kolmogorov Equations......Page 110
3.2. Uniqueness Criterion and Applications......Page 115
3.3. Some Lemmas......Page 126
3.4. Proof of Uniqueness Criterion......Page 128
3.5. Notes......Page 132
4.1. Weak Convergence......Page 133
4.2. General Results......Page 137
4.3. Markov Chains: Time-discrete Case......Page 143
4.4. Markov Chains: Time-continuous Case......Page 152
4.5. Single Birth Processes......Page 164
4.6. Invariant Measures......Page 179
4.7. Notes......Page 184
5.1. Minimum Lp-Metric......Page 186
5.2. Marginality and Regularity......Page 197
5.3. Successful Coupling and Ergodicity......Page 208
5.4. Optimal Markovian Couplings......Page 216
5.5. Monotonicity......Page 223
5.6. Examples......Page 229
5.7. Notes......Page 236
Part II. Symmetrizable Jump Processes......Page 238
6.1. Reversible Markov Processes......Page 240
6.2. Existence......Page 242
6.4. General Representation of Jump Processes......Page 246
6.5. Existence of Honest Reversible Jump Processes......Page 256
6.6. Uniqueness Criteria......Page 262
6.7. Basic Dirichlet Form......Page 268
6.8. Regularity, Extension and Uniqueness......Page 278
6.9. Notes......Page 283
7.1. Field Theory......Page 285
7.2. Lattice Field......Page 289
7.3. Electric Field......Page 293
7.4. Transience of Symmetrizable Markov Chains......Page 297
7.5. Random Walk on Lattice Fractals......Page 311
7.6. A Comparison Theorem......Page 313
7.7. Notes......Page 315
8.1. Introduction to Large Deviations......Page 316
8.2. Rate Function......Page 324
8.3. Upper Estimates......Page 333
8.4. Notes......Page 342
9.1. General Case: an Equivalence......Page 343
9.2. Coupling and Distance Method......Page 353
9.3. Birth-Death Processes......Page 361
9.4. Splitting Procedure and Existence Criterion......Page 372
9.5. Cheeger's Approach and Isoperimetric Constants......Page 381
9.6. Notes......Page 393
Part III. Equilibrium Particle Systems......Page 394
10.1. Introduction......Page 396
10.2. Existence......Page 400
10.3. Uniqueness......Page 404
10.4. Phase Transition: Peierls Method......Page 410
10.5. Ising Model on Lattice Fractals......Page 412
10.6. Reflection Positivity and Phase Transitions......Page 419
10.7. Proof of the Chess-Board Estimates......Page 429
10.8. Notes......Page 434
11.1. Potentiality for Some Speed Functions......Page 435
11.2. Constructions of Gibbs States......Page 438
11.3. Criteria for Reversibility......Page 445
11.4. Notes......Page 459
12.1. Background......Page 460
12.2. Spin Processes from Yang-Mills Lattice Fields......Page 461
12.3. Diffusion Processes from Yang-Mills Lattice Fields......Page 470
12.4. Notes......Page 479
Part IV. Non-equilibrium Particle Systems......Page 480
13.1. Existence Theorems for the Processes......Page 482
13.2. Existence Theorem for Reaction-Diffusion Processes......Page 499
13.3. Uniqueness Theorems for the Processes......Page 506
13.4. Examples......Page 515
13.5. Appendix......Page 523
13.6. Notes......Page 526
14.1. General Results......Page 527
14.2. Ergodicity for Polynomial Model......Page 534
14.3. Reversible Reaction-Diffusion Processes......Page 545
14.4. Notes......Page 551
15.1. Duality......Page 552
15.2. Linear Growth Model......Page 555
15.3. Reaction-Diffusion Processes with Absorbing State......Page 560
15.4. Mean Field Method......Page 563
15.5. Notes......Page 567
16.1. Introduction: Main Results......Page 568
16.2. Preliminaries......Page 572
16.3. Proof of Theorem 16.1......Page 577
16.4. Proof of Theorem 16.3......Page 583
16.5. Notes......Page 584
Bibliography......Page 585
Author Index......Page 602
Subject Index......Page 606
