Green, Brown, and probability & Brownian motion on the line. 2nd revised and augmented ed. (English) Zbl 1014.60001

Singapore: World Scientific (ISBN 981-02-4690-0/pbk; 981-02-4689-7/hbk; 978-981-277-846-8/ebook). x, 170 p. (2002).
This book consists of three parts. Part I constitutes the second edition, revised and augmented, of the author’s book “Green, Brown, and probability” (Singapore, 1995). The author presents from a historical perspective the sound relations between the Brownian motion in probability theory and several fundamental propositions in mathematical physics. In this edition, two sections on convergence at boundary and Green symmetry were added.
Part II of the book is based on the short course “Brownian motion on the line” given by the author at Stanford University [see J. Math. Res. Expo. 1, 74-82 (1981; Zbl 0583.60074) and 2, 87-98 (1982; Zbl 0583.60075)]. It deals with the exit and return properties of the one-dimensional Brownian motion (also with drift) and probabilistic solutions of the Dirichlet and Poisson problems. Finally, the Feynman-Kac formula is discussed. This part of the book contains 27 exercises.
The last and very short Part III “Stopped Feynman-Kac functionals” is a reprint of [the author, in: Séminaire de probabilités XIV. Lect. Notes Math. 784, 347-356 (1980; Zbl 0444.60061)].


60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60J60 Diffusion processes
60J65 Brownian motion
60G17 Sample path properties
60G44 Martingales with continuous parameter
60J35 Transition functions, generators and resolvents
60J45 Probabilistic potential theory
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
47D07 Markov semigroups and applications to diffusion processes
47D08 Schrödinger and Feynman-Kac semigroups
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