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Classical Potential Theory and Its Probabilistic Counterpart

  • J. L. Doob

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 262)

Table of contents

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

  3. Probabilistic Counterpart of Part 1

    1. Front Matter
      Pages 385-385
    2. J. L. Doob
      Pages 387-412
    3. J. L. Doob
      Pages 413-431
    4. J. L. Doob
      Pages 432-462
    5. J. L. Doob
      Pages 539-569
    6. J. L. Doob
      Pages 570-598
    7. J. L. Doob
      Pages 599-626
    8. J. L. Doob
      Pages 627-667
    9. J. L. Doob
      Pages 668-702
  4. Part 3

    1. Front Matter
      Pages 703-703
    2. J. L. Doob
      Pages 719-726
    3. J. L. Doob
      Pages 727-738
  5. Back Matter
    Pages 739-847

About this book

Introduction

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.

Keywords

Markov process Martingale Motion Potential theory Probability theory Transition function

Authors and affiliations

  • J. L. Doob
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-5208-5
  • Copyright Information Springer-Verlag New York 1984
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-9738-3
  • Online ISBN 978-1-4612-5208-5
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site