| 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 |
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