Symmetric Markov Processes, Time Change, and Boundary Theory
The title of this monograph, written by two worldwide known experts on the field, gives a clear idea of the contents of the book, devoted to Markov processes. As the reader can immediately understand from the Preface, the work can be divided in two different parts. The first one (chapters 1,2,3 & 4) contains the definitions of the theory of symmetric Markovian and Dirichlet forms together with many examples and properties. The approach is both probabilistic and analytic and the exposition is rigorous and complete. This part is closely related to other books written by the second authors (in collaboration with others) as for example “Dirichlet forms and symmetric Markov processes by Fukushima”, de Gruyter Stud. Math. 19, Berlin, 1994.
The second part (chapters 5, 6 &7) contains some recent results on time changes and boundary theory, topics which are first presented here in the format of a book. In particular, with respect to time changes, a characterization of time-changed Markov process in terms of Douglas integrals is presented. On the other hand, the boundary theory studies the problem of extending a Markov process from its original space to a larger one in which the original is an open subset and such that the new process spends zero time in the extension.
The presentation is comprehensive and self contained though it requires solid previous knowledge on theoretic probability together with some notions on measure theory and functional analysis. The volume can serve as textbook for graduate students or as a reference for researchers on the field.