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  • Book
  • © 1984

Classical Potential Theory and Its Probabilistic Counterpart

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Authors:

  1. J. L. Doob

    1. Department of Mathematics, University of Illinois, Urbana, USA

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

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Table of contents (32 chapters)

  1. Front Matter

    Pages i-xxv
    PDF

  2. Classical and Parabolic Potential Theory

    1. Front Matter

      Pages 1-1
      PDF

    5. Potentials on Special Open Sets

      • J. L. Doob
      Pages 45-56

    6. Polar Sets and Their Applications

      • J. L. Doob
      Pages 57-69

    8. Green Functions

      • J. L. Doob
      Pages 85-97

    11. The Sweeping Operation

      • J. L. Doob
      Pages 155-165

    12. The Fine Topology

      • J. L. Doob
      Pages 166-194

    13. The Martin Boundary

      • J. L. Doob
      Pages 195-225

    14. Classical Energy and Capacity

      • J. L. Doob
      Pages 226-255

    15. One-Dimensional Potential Theory

      • J. L. Doob
      Pages 256-261

    16. Parabolic Potential Theory: Basic Facts

      • J. L. Doob
      Pages 262-284

    18. Parabolic Potential Theory (Continued)

      • J. L. Doob
      Pages 295-328
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About this book

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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Keywords

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Authors and Affiliations

  • Department of Mathematics, University of Illinois, Urbana, USA

    J. L. Doob

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Bibliographic Information

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Buy it now

eBook USD 74.99
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  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

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