The book under review is the fourth volume in a series of five books by the authors: Chapter I-IV (1975; Zbl 0323.60039) contain the theory of analytic sets, capacities and applications to stochastic processes, Chapter V-VIII (1980; Zbl 0464.60001) contain martingale theory and Chapter IX-XI (1983; Zbl 0526.60001) contain analytic potential theory of single kernels.
In the first part of this volume, the authors continue the analytic potential theory of the previous volume in the continuous time cases. In the latter half, general theory of Markov processes and results concerning additive and multiplicative functionals are treated. Concerning these topics, this book contains extensive results including new ones and the proofs are elegant.
In Chapter XII, some potential principles and ergodic theorems for resolvents and exit laws are treated. For a pair of excessive measure and excessive function, the “mass” of it is defined and its properties are studied. Ray resolvents and Ray-Knight compactification are treated in this chapter. In the first part of Chapter XIII, Hille-Yosida’s theorem on the correspondences of semigroups, resolvents and generators, and Hung’s theorem on construction of resolvents from potential operators are treated. Typical examples of semigroups and resolvents are given in this chapter. Energy (Dirichlet) forms of not necessarily symmetric resolvents are concerned in the last part of this chapter.
Chapter XIV contains the general concepts of Markov processes; construction of Markov processes from Feller and Ray semigroups, time reversions moderate processes, strong Markov processes and exceptional sets.
Chapter XV is devoted to the study of additive functionals and its applications. The domain of the extended generator is defined and characterizations by the oblique bracket of the associated additive functional are discussed. Lévy systems of Markov processes and their relations with generators are treated. The local time of a random set is defined as an additive functional supported by the closure of the set. Concerning the local time of a fixed point, Itô’s theorem on Poisson point processes of excursions is explained.
In connection with the transformation of Markov processes by multiplicative functionals, the notion of right process is necessary since the transformed process is a right process in general. Detailed presentations of this are offered at the beginning of Chapter XVI. General and some special transformations by multiplicative functionals such as h-transformations of Doob and subprocesses by terminal times are treated in this chapter.
If one reads carefully the book under review, one will find that it contains many interesting concepts including new notions. The authors say that the last three chapters on Markov processes are presented for the preparation of the forthcoming last volume of this series.