Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
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Biography of Joseph L. Doob
Born in Cincinnati, Ohio on February 27, 1910, Joseph L. Doob studied for both his undergraduate and doctoral degrees at Harvard University. He was appointed to the University of Illinois in 1935 and remained there until his retirement in 1978.
Doob worked first in complex variables, then moved to probability under the initial impulse of H. Hotelling, and influenced by A.N Kolmogorov's famous monograph of 1933, as well as by Paul Lévy's work.
In his own book Stochastic Processes (1953), Doob established martingales as a particularly important type of stochastic process. Kakutani's treatment of the Dirichlet problem in 1944, combining complex variable theory and probability, sparked off Doob's interest in potential theory, which culminated in the present book.
(For more details see: http://www.dartmouth.edu/~chance/Doob/conversation.html)Review:
From the reviews:
"In the early 1920's, Norbert Wiener wrote significant papers on the Dirichlet problem and on Brownian motion. Since then there has been enormous activity in potential theory and stochastic processes, in which both subjects have reached a high degree of polish and their close relation has been discovered. Here is a momumental work by Doob, one of the masters, in which Part 1 develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part 1". G.E.H. Reuter in Short Book Reviews (1985)
"This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fullfilled in a masterly manner". Metrika (1986)
"It is good news that Doob’s monumental book is now available at a very reasonable price. The impressive volume (846 pages!) is still the only book concentrating on a thorough presentation of the potential theory of the Laplace operator ... . The material in the chapters on conditional Brownian motion and Brownian motion on the Martin space cannot easily be found in that depth elsewhere. A long appendix on various topics (more than 50 pages) and many historical notes complete this great ‘encyclopedia’." (Wolfhard Hansen, Zentralblatt MATH, Vol. 990 (15), 2002)
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Book Description Secaucus, New Jersey, U.S.A.: Springer Verlag, 1984. Soft cover. Book Condition: New. Dust Jacket Condition: New. 1st Edition. **INTERNATIONAL EDITION** Read carefully before purchase: This book is the international edition in mint condition with the different ISBN and book cover design, the major content is printed in full English as same as the original North American edition. The book printed in black and white, generally send in twenty-four hours after the order confirmed. All shipments go through via USPS/UPS/DHL with tracking numbers. Great professional textbook selling experience and expedite shipping service. Bookseller Inventory # ABE-10833602265
Book Description Secaucus, New Jersey, U.S.A.: Springer Verlag, 1984. Soft cover. Book Condition: New. Dust Jacket Condition: New. 1st Edition. Low price guarantee! The book is the brand new international edition textbook with the different ISBN and cover design. The book main content black/white printed in full English as same as the corresponding original US edition. Fast shipments in one to two working days after the orders confirmed. Bookseller Inventory # ABE-10833602263
Book Description Springer, 1984. Hardcover. Book Condition: New. Never used!. Bookseller Inventory # P110387908811
Book Description Springer. Hardcover. Book Condition: New. 0387908811 New Condition. Bookseller Inventory # NEW7.0127890