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© 2001

Classical Potential Theory and Its Probabilistic Counterpart

Book

Part of the Classics in Mathematics book series (CLASSICS)

Table of contents

  1. Front Matter
    Pages N1-xxv
  2. Classical and Parabolic Potential Theory

    1. Front Matter
      Pages 1-1
    2. Joseph L. Doob
      Pages 45-56
    3. Joseph L. Doob
      Pages 57-69
    4. Joseph L. Doob
      Pages 85-97
    5. Joseph L. Doob
      Pages 141-154
    6. Joseph L. Doob
      Pages 155-165
    7. Joseph L. Doob
      Pages 166-194
    8. Joseph L. Doob
      Pages 195-225
    9. Joseph L. Doob
      Pages 226-255
    10. Joseph L. Doob
      Pages 256-261
    11. Joseph L. Doob
      Pages 262-284
    12. Joseph L. Doob
      Pages 295-328

About this book

Introduction

From the reviews: "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 fulfilled in a masterly manner".
M. Brelot in Metrika (1986)

Keywords

31XX Brownian motion Markov process Martingale Potential theory Probabilistic Potential Theory Stochastic processes Uniform integrability measure theory stochastic process

Authors and affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

About the authors

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)

Bibliographic information

  • Book Title Classical Potential Theory and Its Probabilistic Counterpart
  • Authors Joseph L. Doob
  • Series Title Classics in Mathematics
  • DOI https://doi.org/10.1007/978-3-642-56573-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2001
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-41206-9
  • eBook ISBN 978-3-642-56573-1
  • Series ISSN 1431-0821
  • Edition Number 1
  • Number of Pages XXIV, 846
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published as Vol. 262 in the series: Grundlehren der mathematischen Wissenschaften
  • Topics Potential Theory
    Probability Theory and Stochastic Processes
  • Buy this book on publisher's site

Reviews

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)