##
**Diffusions, Markov processes, and martingales. Vol. 1: Foundations.
2nd ed.**
*(English)*
Zbl 0826.60002

Chichester: Wiley. xx, 386 p. (1994).

This second edition is a considerably expanded version of the first one (1979; Zbl 0402.60003). The number of pages increased by more than 60 per cent, the number of authors even by 100 per cent. Especially the first two of the three chapters (“Brownian motion” and “Some classical theory”) have undergone changes and extensions while the third chapter on Markov processes remains basically as it was. In spite of the changes (and the omission of the musical allusions) the book retains the very lively style of the first edition with lots of explanation, remarks, examples, warnings and references.

The first chapter contains a wealth of material on Brownian motion, including some rather recent results. The chapter introduces many concepts in the context of Brownian motion which are developed more thoroughly later in the book, e.g. martingales, Gaussian processes, Markov processes and Lévy processes. Donsker’s theorem is proved via Skorokhod embedding. A study of sample-path properties, local time, Strassen’s LIL, connection to potential theory and winding numbers follows. The emphasis is put more on conveying an understanding of the subject than on complete proofs (McKean’s proof of the LIL for Brownian motion is provided but that of Strassen’s LIL is not).

Chapter 2 – which can be read before Chapter 1 – starts with subchapters on basic measure theory, basic probability theory and stochastic processes which are much more detailed than in the first edition. The section on discrete-parameter martingales is inspired by the second author’s book “Probability with martingales” (1991; Zbl 0722.60001) but more condensed. The next section on continuous-parameter supermartingales contains a paragraph on début and section theorems. A subchapter on probability measures on Lusin spaces has been added.

The third chapter on Markov processes with subchapters on the Hille- Yosida theorem, Feller-Dynkin processes, additive functionals, Martin boundary and Ray processes has remained largely unchanged since – as the authors rightly state in the preface – it has been regarded as the most successful part of the first edition. Indeed the treatment of Martin boundary and Ray processes remains the most readable one still today.

[Vol. 2 has been published in 1987 (Zbl 0627.60001)].

The first chapter contains a wealth of material on Brownian motion, including some rather recent results. The chapter introduces many concepts in the context of Brownian motion which are developed more thoroughly later in the book, e.g. martingales, Gaussian processes, Markov processes and Lévy processes. Donsker’s theorem is proved via Skorokhod embedding. A study of sample-path properties, local time, Strassen’s LIL, connection to potential theory and winding numbers follows. The emphasis is put more on conveying an understanding of the subject than on complete proofs (McKean’s proof of the LIL for Brownian motion is provided but that of Strassen’s LIL is not).

Chapter 2 – which can be read before Chapter 1 – starts with subchapters on basic measure theory, basic probability theory and stochastic processes which are much more detailed than in the first edition. The section on discrete-parameter martingales is inspired by the second author’s book “Probability with martingales” (1991; Zbl 0722.60001) but more condensed. The next section on continuous-parameter supermartingales contains a paragraph on début and section theorems. A subchapter on probability measures on Lusin spaces has been added.

The third chapter on Markov processes with subchapters on the Hille- Yosida theorem, Feller-Dynkin processes, additive functionals, Martin boundary and Ray processes has remained largely unchanged since – as the authors rightly state in the preface – it has been regarded as the most successful part of the first edition. Indeed the treatment of Martin boundary and Ray processes remains the most readable one still today.

[Vol. 2 has been published in 1987 (Zbl 0627.60001)].

Reviewer: M.Scheutzow (Berlin)

### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60Jxx | Markov processes |

60G44 | Martingales with continuous parameter |

60G17 | Sample path properties |

60J55 | Local time and additive functionals |