This book evolved from several stacks of lecture notes written over a decade and given in classes at slightly varying levels. In transforming the over lapping material into a book, I aimed at presenting some of the best features of the subject with a minimum of prerequisities and technicalities. (Needless to say, one man's technicality is another's professionalism. ) But a text frozen in print does not allow for the latitude of the classroom; and the tendency to expand becomes harder to curb without the constraints of time and audience. The result is that this volume contains more topics and details than I had intended, but I hope the forest is still visible with the trees. The book begins at the beginning with the Markov property, followed quickly by the introduction of option al times and martingales. These three topics in the discrete parameter setting are fully discussed in my book A Course In Probability Theory (second edition, Academic Press, 1974). The latter will be referred to throughout this book as the Course, and may be considered as a general background; its specific use is limited to the mate rial on discrete parameter martingale theory cited in § 1. 4. Apart from this and some dispensable references to Markov chains as examples, the book is self-contained.
From the reviews of the First Edition:
This excellent book is based on several sets of lecture notes written over a decade and has its origin in a one-semester course given by the author at the ETH, Zürich, in the spring of 1970. The author's aim was to present some of the best features of Markov processes and, in particular, of Brownian motion with a minimum of prerequisites and technicalities. The reader who becomes acquainted with the volume cannot but agree with the reviewer that the author was very successful in accomplishing this goal...The volume is very useful for people who wish to learn Markov processes but it seems to the reviewer that it is also of great interest to specialists in this area who could derive much stimulus from it. One can be convinced that it will receive wide circulation.
H.J. Engelbert, MathSciNet
From the reviews of the second edition:
"This monograph is a considerably extended second edition of K.L. Chung’s classic ‘Lectures from Markov processes to Brownian motion’ ... . Adding to Chung’s masterpiece is a formidable task; the new chapters by Walsh capture the spirit of the original and give a gentle, inspiring and eminently useful introduction to Ray processes, time reversal and duality." (René L. Schilling, Zentralblatt MATH, Vol. 1082, 2006)
"The volume under review is the union of two distinct, albeit complementary, works. ... there are a number of interesting examples in the more familiar world of Markov chains which the reader can use to gain insight into the new ideas. ... If I were asked to recommend a book to beginners who wished to immerse themselves in the subject and emerge in a fit state to tackle the contemporary literature, then I would choose this volume without hesitation." (David Applebaum, The Mathematical Gazette, Vol. 92 (523), 2008)
