简介
This volume presents a representative work of Chinese probabilists on probability theory and its applications in physics. Interesting results of jump Markov processes are discussed, as well as Markov interacting processes with noncompact states, including the Schlogal model taken from statistical physics. The main body of this book is self-contained and can be used in a course in stochastic processes for graduate students. The book consists of four parts. In Parts 1 and 2, the author introduces the general theory for jump processes. New contributions to the classical problems: uniqueness, recurrence and positive recurrence are studied. Then, probability metrics and coupling methods, stochastically monotonicity, reversibility, large deviations and the estimates of L squared-spectral gap are discussed. Part 3 begins with the study of equilibrium particle systems. This contains the criteria of the reversibility, the construction of Gibbs states and the particle systems on lattice fractals. The final part emphasizes the reaction-diffusion processes which come from non-equilibrium statistical physics. Topics include constructions, existence of stationary distributions, ergodicity, phase transitions and hydrodynamic limits for the processes.
目录
Starting from Markov Chains. An Overview of the Book p. 1
Three Classical Problems for Markov Chains p. 1
Probability Metrics and Coupling Methods p. 6
Reversible Markov Chains p. 13
Large Deviations and Spectral Gap p. 14
Equilibrium Particle Systems p. 16
Non-Equilibrium Particle Systems p. 18
General Jump Processes
Transition Function and its Laplace Transform p. 23
Basic Properties of Transition Function p. 23
q-Pair p. 27
Differentiability p. 38
Laplace Transforms p. 51
Notes p. 57
Existence and Simple Constructions of Jump Processes p. 62
Non-Negative Solutions p. 62
Kolmogorov Equations and Minimal Jump Process p. 69
Some Sufficient Conditions for Uniqueness p. 79
Kolmogorov Equations and q-Condition p. 84
Entrance Space and Exit Space p. 87
Construction of q-Processes with Single-Exit q-Pair p. 92
Notes p. 95
Uniqueness Criteria p. 97
Uniqueness Criteria Related to Kolmogorov Equations p. 97
Uniqueness Criterion and Applications p. 102
Some Lemmas p. 112
Proof of Uniqueness Criterion p. 115
Notes p. 119
Recurrence, Ergodicity and Invariant Measures p. 120
Weak Convergence p. 120
General Results p. 124
Markov Chains. Time-Discrete Case p. 130
Markov Chains. Time-Continuous Case p. 140
Single Birth Processes p. 152
Invariant measures p. 160
Notes p. 165
Probability Metrics and Coupling Methods p. 167
Minimum L[actual symbols not reproducible]-Metric p. 167
Marginality and Regularity p. 174
Successful Coupling p. 181
Monotonicity p. 187
Examples p. 194
Notes p. 200
Symmetrizable Jump Processes
Symmetrizable Jump Processes and Dirichlet Forms p. 207
Reversible Markov processes p. 207
Existence p. 210
Equivalence of Backward and Forward Kolmogorov Equations p. 213
General Representation of Jump Processes p. 213
Existence of Honest Reversible Jump Processes p. 223
Uniqueness Criteria p. 230
Basic Dirichlet Form p. 235
Regularity, Extension and Uniqueness p. 245
Notes p. 250
Field Theory p. 252
Field Theory p. 252
Lattice Field p. 256
Electric Field p. 260
Transience of Symmetrizable Markov Chains p. 264
Random Work on Lattice Fractals p. 278
A Comparison Theorem p. 281
Notes p. 283
Large Deviations p. 284
Introduction to Large Deviations p. 284
Rate Function p. 292
Upper Estimates p. 302
Notes p. 310
Spectral Gap p. 311
General Case p. 311
Reversible Case p. 317
Markov Chains p. 321
Notes p. 332
Equilibrium Particle Systems
Random Fields p. 339
Introduction p. 339
Existence p. 343
Uniqueness p. 347
Phase Transition, Peierls Method p. 353
Ising Model on Lattice Fractals p. 356
Reflection Positivity and Phase Transitions p. 362
Proof of the Chess-Board Estimates p. 372
Notes p. 377
Reversible Spin Processes and Exclusion Processes p. 379
Potentiality for Some Speed Functions p. 379
Constructions of Gibbs States p. 382
Criteria for Reversibility p. 389
Notes p. 403
Yang-Mills Lattice Fields p. 404
Background p. 404
Spin Processes From Yang-Mills Lattice Fields p. 405
Diffusion Processes From Yang-Mills Lattice Fields p. 414
Notes p. 423
Non-Equilibrium Particle Systems
Constructions of the Processes p. 427
Existence Theorems for the Processes p. 427
Existence Theorem for Reaction-Diffusion Processes p. 444
Uniqueness Theorems for the Processes p. 451
Examples p. 460
Notes p. 468
Existence of the Stationary Distributions and Ergodicity p. 472
General Results p. 472
u-Criterion for Ergodicity p. 479
Applications p. 489
Reversible Reaction-Diffusion Processes p. 496
Notes p. 501
Phase Transitions p. 502
Duality p. 502
Linear Growth Model p. 505
Reaction-diffusion Processes with Absorbing State p. 510
Mean Field Method p. 513
Notes p. 517
Hydrodynamic Limits p. 518
Introduction. Main Results p. 518
Preliminaries p. 520
Proof of Theorem 16.1 p. 525
Proof of Theorem 16.3 p. 531
Notes p. 532
Bibliography p. 534
Index p. 544
Three Classical Problems for Markov Chains p. 1
Probability Metrics and Coupling Methods p. 6
Reversible Markov Chains p. 13
Large Deviations and Spectral Gap p. 14
Equilibrium Particle Systems p. 16
Non-Equilibrium Particle Systems p. 18
General Jump Processes
Transition Function and its Laplace Transform p. 23
Basic Properties of Transition Function p. 23
q-Pair p. 27
Differentiability p. 38
Laplace Transforms p. 51
Notes p. 57
Existence and Simple Constructions of Jump Processes p. 62
Non-Negative Solutions p. 62
Kolmogorov Equations and Minimal Jump Process p. 69
Some Sufficient Conditions for Uniqueness p. 79
Kolmogorov Equations and q-Condition p. 84
Entrance Space and Exit Space p. 87
Construction of q-Processes with Single-Exit q-Pair p. 92
Notes p. 95
Uniqueness Criteria p. 97
Uniqueness Criteria Related to Kolmogorov Equations p. 97
Uniqueness Criterion and Applications p. 102
Some Lemmas p. 112
Proof of Uniqueness Criterion p. 115
Notes p. 119
Recurrence, Ergodicity and Invariant Measures p. 120
Weak Convergence p. 120
General Results p. 124
Markov Chains. Time-Discrete Case p. 130
Markov Chains. Time-Continuous Case p. 140
Single Birth Processes p. 152
Invariant measures p. 160
Notes p. 165
Probability Metrics and Coupling Methods p. 167
Minimum L[actual symbols not reproducible]-Metric p. 167
Marginality and Regularity p. 174
Successful Coupling p. 181
Monotonicity p. 187
Examples p. 194
Notes p. 200
Symmetrizable Jump Processes
Symmetrizable Jump Processes and Dirichlet Forms p. 207
Reversible Markov processes p. 207
Existence p. 210
Equivalence of Backward and Forward Kolmogorov Equations p. 213
General Representation of Jump Processes p. 213
Existence of Honest Reversible Jump Processes p. 223
Uniqueness Criteria p. 230
Basic Dirichlet Form p. 235
Regularity, Extension and Uniqueness p. 245
Notes p. 250
Field Theory p. 252
Field Theory p. 252
Lattice Field p. 256
Electric Field p. 260
Transience of Symmetrizable Markov Chains p. 264
Random Work on Lattice Fractals p. 278
A Comparison Theorem p. 281
Notes p. 283
Large Deviations p. 284
Introduction to Large Deviations p. 284
Rate Function p. 292
Upper Estimates p. 302
Notes p. 310
Spectral Gap p. 311
General Case p. 311
Reversible Case p. 317
Markov Chains p. 321
Notes p. 332
Equilibrium Particle Systems
Random Fields p. 339
Introduction p. 339
Existence p. 343
Uniqueness p. 347
Phase Transition, Peierls Method p. 353
Ising Model on Lattice Fractals p. 356
Reflection Positivity and Phase Transitions p. 362
Proof of the Chess-Board Estimates p. 372
Notes p. 377
Reversible Spin Processes and Exclusion Processes p. 379
Potentiality for Some Speed Functions p. 379
Constructions of Gibbs States p. 382
Criteria for Reversibility p. 389
Notes p. 403
Yang-Mills Lattice Fields p. 404
Background p. 404
Spin Processes From Yang-Mills Lattice Fields p. 405
Diffusion Processes From Yang-Mills Lattice Fields p. 414
Notes p. 423
Non-Equilibrium Particle Systems
Constructions of the Processes p. 427
Existence Theorems for the Processes p. 427
Existence Theorem for Reaction-Diffusion Processes p. 444
Uniqueness Theorems for the Processes p. 451
Examples p. 460
Notes p. 468
Existence of the Stationary Distributions and Ergodicity p. 472
General Results p. 472
u-Criterion for Ergodicity p. 479
Applications p. 489
Reversible Reaction-Diffusion Processes p. 496
Notes p. 501
Phase Transitions p. 502
Duality p. 502
Linear Growth Model p. 505
Reaction-diffusion Processes with Absorbing State p. 510
Mean Field Method p. 513
Notes p. 517
Hydrodynamic Limits p. 518
Introduction. Main Results p. 518
Preliminaries p. 520
Proof of Theorem 16.1 p. 525
Proof of Theorem 16.3 p. 531
Notes p. 532
Bibliography p. 534
Index p. 544
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