The development of mathematics showed that the following three branches, functional analysis, partial differential equations and diffusion Markov processes, are interconnected very strongly. The roots of these connections go back to the fundamental results of A. N. Kolmogorov obtained at the beginning of thirties. Essential contributions in this field are due to W. Feller, P. Levy, M. Kac, E. Dynkin, K. L. Chung, M. Freidlin, N. Krylov and others. The book under review provides the readers the rare opportunity to get knowledge and to learn effectively this circle of topics. The book is a result of graduate courses delivered by the author at several universities.
The book starts with a chapter called “Introductory and Summary”. Here the author describes important examples of diffusion Markov processes and shows how they can be studied by the methods of functional analysis and using partial differential equations. According to the subjects, the basis material of the book is divided into five parts. Let us describe them shortly.
The first part (Chapters 1-4) contains the basic facts of the Lebesgue theory of measures and integration, manifold theory, functional analysis and distribution theory. This material, included here for a completeness and reader’s convenience, is used essentially in the whole book.
Chapters 5-6 form the second part of the book. Here the reader will find basic results about Sobolev spaces, pseudo-differential operators and topics from the modern potential theory.
The third part (Chapter 7) is devoted to various maximum principles for degenerate elliptic differential operators. The importance of these topics for studying Markov process is demonstrated in Chapter 10.
The fourth part (Chapter 8) deals with general boundary value problems for second order elliptic differential operators. The basic questions, such as existence, uniqueness and regularity of solutions of general boundary value problems with spectral parameter are studied in the framework of Sobolev spaces using the calculus of pseudo-differential operators. These results are also very useful for constructing Markov processes.
The fifth part (Chapters 9-10) is devoted to the functional analytical approach to the problem of construction of Markov processes. Very general existence theorems for Markov processes are proved in Chapter 9. Further, in Chapter 10 the author describes the construction of Markov processes by solving boundary value problems with spectral parameter.
Let us mention that useful bibliographical references are indicated primarily in Notes at the end of each chapter. Naturally, the book ends with Bibliography, List of Symbols and Index of terms.
Thus, the well chosen material given in a suitable form and style makes the book very useful for last year university students as well as for professional mathematicians with interests in at least one of the fields: probability theory, functional analysis and partial differential equations.