9789812388117

From Markov Chains to Non-Equilibrium Particle Systems

by
  • ISBN13:

    9789812388117

  • ISBN10:

    9812388117

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2004-03-01
  • Publisher: World Scientific Pub Co Inc
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Supplemental Materials

What is included with this book?

Summary

- A Summary of the research results of outstanding probabilists and statistics is given including Professor Zhenting Hou, who was awarded with the D Anderson's prize from Cambridge in the 1980s; Professor D W Stroock, a vice president of Chinese Mathematical Society and an academician in USA; and Professor T M Liggett, one of the leaders in the field.- Markov chains is a popular subject in UK, Australia, Russia etc.- The interacting particle systems is an active research area in the world now, including USA, Russia, Germany, France, Italy, Japan and may others.

Table of Contents

Preface to the First Edition ix
Preface to the Second Edition xi
An Overview of the Book: Starting from Markov Chains
1(20)
Three Classical Problems for Markov Chains
1(5)
Probability Metrics and Coupling Methods
6(7)
Reversible Markov Chains
13(2)
Large Deviations and Spectral Gap
15(2)
Equilibrium Particle Systems
17(2)
Non-equilibrium Particle Systems
19(2)
Part I. General Jump Processes
21(204)
Transition Function and its Laplace Transform
23(39)
Basic Properties of Transition Function
23(4)
The q-Pair
27(11)
Differentiability
38(13)
Laplace Transforms
51(6)
Appendix
57(4)
Notes
61(1)
Existence and Simple Constructions of Jump Processes
62(35)
Minimal Nonnegative Solutions
62(8)
Kolmogorov Equations and Minimal Jump Process
70(9)
Some Sufficient Conditions for Uniqueness
79(6)
Kolmogorov Equations and q-Condition
85(3)
Entrance Space and Exit Space
88(5)
Construction of q-Processes with Single-Exit q-Pair
93(3)
Notes
96(1)
Uniqueness Criteria
97(23)
Uniqueness Criteria Based on Kolmogorov Equations
97(5)
Uniqueness Criterion and Applications
102(11)
Some Lemmas
113(2)
Proof of Uniqueness Criterion
115(4)
Notes
119(1)
Recurrence, Ergodicity and Invariant Measures
120(53)
Weak Convergence
120(4)
General Results
124(6)
Markov Chains: Time-discrete Case
130(9)
Markov Chains: Time-continuous Case
139(12)
Single Birth Processes
151(15)
Invariant Measures
166(5)
Notes
171(2)
Probability Metrics and Coupling Methods
173(52)
Minimum Lp-Metric
173(11)
Marginality and Regularity
184(11)
Successful Coupling and Ergodicity
195(8)
Optimal Markovian Couplings
203(7)
Monotonicity
210(6)
Examples
216(7)
Notes
223(2)
Part II. Symmetrizable Jump Processes
225(156)
Symmetrizable Jump Processes and Dirichlet Forms
227(45)
Reversible Markov Processes
227(2)
Existence
229(4)
Equivalence of Backward and Forward Kolmogorov Equations
233(1)
General Representation of Jump Processes
233(10)
Existence of Honest Reversible Jump Processes
243(6)
Uniqueness Criteria
249(6)
Basic Dirichlet Form
255(10)
Regularity, Extension and Uniqueness
265(5)
Notes
270(2)
Field Theory
272(31)
Field Theory
272(4)
Lattice Field
276(4)
Electric Field
280(4)
Transience of Symmetrizable Markov Chains
284(14)
Random Walk on Lattice Fractals
298(2)
A Comparison Theorem
300(2)
Notes
302(1)
Large Deviations
303(27)
Introduction to Large Deviations
303(8)
Rate Function
311(9)
Upper Estimates
320(9)
Notes
329(1)
Spectral Gap
330(51)
General Case: an Equivalence
330(10)
Coupling and Distance Method
340(8)
Birth-Death Processes
348(11)
Splitting Procedure and Existence Criterion
359(9)
Cheeger's Approach and Isoperimetric Constants
368(12)
Notes
380(1)
Part III. Equilibrium Particle Systems
381(86)
Random Fields
383(39)
Introduction
383(4)
Existence
387(4)
Uniqueness
391(6)
Phase Transition: Peierls Method
397(2)
Ising Model on Lattice Fractals
399(7)
Reflection Positivity and Phase Transitions
406(10)
Proof of the Chess-Board Estimates
416(5)
Notes
421(1)
Reversible Spin Processes and Exclusion Processes
422(25)
Potentiality for Some Speed Functions
422(3)
Constructions of Gibbs States
425(7)
Criteria for Reversibility
432(14)
Notes
446(1)
Yang-Mills Lattice Field
447(20)
Background
447(1)
Spin Processes from Yang-Mills Lattice Fields
448(9)
Diffusion Processes from Yang-Mills Lattice Fields
457(9)
Notes
466(1)
Part IV. Non-equilibrium Particle Systems
467(105)
Constructions of the Processes
469(45)
Existence Theorems for the Processes
469(17)
Existence Theorem for Reaction-Diffusion Processes
486(7)
Uniqueness Theorems for the Processes
493(9)
Examples
502(8)
Appendix
510(3)
Notes
513(1)
Existence of Stationary Distributions and Ergodicity
514(25)
General Results
514(7)
Ergodicity for Polynomial Model
521(11)
Reversible Reaction-Diffusion Processes
532(6)
Notes
538(1)
Phase Transitions
539(16)
Duality
539(3)
Linear Growth Model
542(5)
Reaction-Diffusion Processes with Absorbing State
547(3)
Mean Field Method
550(4)
Notes
554(1)
Hydrodynamic Limits
555(17)
Introduction: Main Results
555(4)
Preliminaries
559(5)
Proof of Theorem 16.1
564(6)
Proof of Theorem 16.3
570(1)
Notes
571(1)
Bibliography 572(17)
Author Index 589(4)
Subject Index 593

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