From Markov Chains to Non-Equilibrium Particle Systems

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.