General theory of Markov processes.

*(English)*Zbl 0649.60079
Pure and Applied Mathematics, 133. Boston, MA: Academic Press, Inc. xi, 419 p. $ 49.50 (1988).

This book is devoted to a deep study of right processes, a class of Markov processes which contains all the reasonable processes and is stable under a large number of probabilistic transformations. It contains the results proved on general Markov processes over a period of nearly twenty years [the well-known book of R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, appeared in 1968 see Zbl 0169.492)] with the notable exceptions of: Martin boundaries, time reversal, and Kuznetsov measures. One should note as special features of the book the importance given to homogeneous random measures (rather than to just additive functionals), a complete discussion of Lévy systems and of excursion theory.

The quality of the text is outstanding, the greatest care being given to every detail. The technical level is rather high, and methods of the so- called “general theory of processes” are used all over the book, as well as the techniques of Ray compactification; but the author has been careful to sketch the results he needs in concise and well written Appendices. This book will obviously become the standard reference in this theory.

Contents: Fundamental hypotheses. Transformations. Homogeneity. Random measures. Ray-Knight (compactification) methods. Stochastic calculus. Multiplicative functionals. Additive functionals. Appendices. Indexes.

The quality of the text is outstanding, the greatest care being given to every detail. The technical level is rather high, and methods of the so- called “general theory of processes” are used all over the book, as well as the techniques of Ray compactification; but the author has been careful to sketch the results he needs in concise and well written Appendices. This book will obviously become the standard reference in this theory.

Contents: Fundamental hypotheses. Transformations. Homogeneity. Random measures. Ray-Knight (compactification) methods. Stochastic calculus. Multiplicative functionals. Additive functionals. Appendices. Indexes.

Reviewer: P.A.Meyer

##### MSC:

60J25 | Continuous-time Markov processes on general state spaces |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60J40 | Right processes |

60J45 | Probabilistic potential theory |