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From Markov chains to non-equilibrium particle systems. (English) Zbl 0753.60055
Singapore: World Scientific. x, 550 p. (1992).
The book is a comprehensive account of the theory of jump processes and particle systems. The author is an outstanding Chinese specialist in probability theory and stochastic processes creating the Chinese school of Markov processes. The material is divided into five parts: (1) Overview and preliminary results: classical Markov chains, coupling ideas, large deviations, particle systems. (2) General jump processes: transition function and Laplace transform, existence and construction of jump processes, uniqueness criteria, recurrence, ergodicity and invariant measures, probability metrics and coupling methods. (3) Symmetrizable jump processes: symmetrizable jump processes and Dirichlet forms, field theory, large deviations, spectral gap. (4) Equilibrium particle systems: random fields, reversible spin processes and exclusion processes, Yang-Mills lattice field. (5) Non-equilibrium particle systems: construction of the processes, existence of the stationary distributions and ergodicity, phase transitions, hydrodynamical limits.
Reviewer’s remark: Probability metrics and related coupling ideas are thoroughly studied in the books of the reviewer [Probability metrics and the stability of stochastic models (1991; Zbl 0744.60004)] and of V. V. Kalashnikov and the reviewer [Mathematical methods for construction of queueing models (1990; Zbl 0709.60096)].

60Jxx Markov processes
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60Axx Foundations of probability theory
60Fxx Limit theorems in probability theory
81T13 Yang-Mills and other gauge theories in quantum field theory
60Exx Distribution theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics