This book is a revised edition of the Chinese original (1980; Zbl 0606.60067) and the fifth volume in the Series in Pure and Applied Mathematics. Its objective is to describe the fundamental theory of birth and death processes and Markov chains and to present the recent results in this field. Under Markov chain here is understood a homogeneous Markov process with continuous time parameter and countable set of states. Chains of this kind are important not only because its comparatively complete and unified theory can be used for reference in general Markov chains and other stochastic processes, but also because there is a steady increase in their applications to natural sciences and practical problems such as physics, biology, chemistry, programming theory and queueing theory.
The book contains eight chapters and two appendices. Chapter 1 provides general concepts of stochastic processes used in the book (definition, separability and measurability of stochastic processes, conditional probabilities and mathematical expectations, the Markov property and transition probabilities). Chapter 2 studies analytic properties of transition probability functions of Markov chains and derives forward and backward equations (general properties and differentiability of standard transition matrices, minimal \(Q\)-resolvent matrices and their properties, exit and entrance families). Chapter 3 deals with properties of sample functions (sets and intervals of constancy, right lower semi-continuity, strong Markov property). Chapter 4 treats zero-one laws, recurrence and an embedding problem for transition probabilities. Chapter 5 gives the basic theory of birth and death processes (probabilistic meanings of numerical characteristics, random functionals of upward integral type, first entrance time and sojourn time, solutions of Kolmogorov equations and stationary distributions, some examples of application of birth and death processes). Chapter 6 deals with the construction theory of birth and death processes, and solves the observe and reverse problems encountered here. The basic idea is similar to the constructionism of functions: the properties of sample functions suggest that the structure of Doob processes is simpler, so it is provided to approach an arbitrary \(Q\)-process. Chapter 7 offers the analytical construction of birth and death processes (numerical characteristics and classification of boundary points, minimal solution, representations for exit and entrance families, recurrence and ergodicity). Appendix 1 draws attention to the close relations between Markov processes and the potential theory in the classical analysis discussing certain aspects of excessive functions. Appendix 2 is devoted to some lemmas from measure theory playing an important role in the theory of stochastic processes; it deals with \(\lambda\)-systems and the \({\mathfrak L}\)-system method.
This monograph is the first systematic treatment of the subject in compact book form showing the actual stage of research in this field. It makes first, for the first time, accessible a lot of original results of the authors and other scientists in English, published earlier in Chinese in separate papers. It is an indispensable book for researchers working in this field and its clear style makes it exceptionally useful for people using birth and death processes and Markov chains in applications.