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Classical Potential Theory and Its Probabilistic Counterpart

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  • Book
  • © 1984

Overview

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
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  2. Classical and Parabolic Potential Theory

    1. Front Matter

      Pages 1-1
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    2. Introduction to the Mathematical Background of Classical Potential Theory

      • J. L. Doob
      Pages 3-13

    3. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions

      • J. L. Doob
      Pages 14-34

    4. Infima of Families of Superharmonic Functions

      • J. L. Doob
      Pages 35-44

    5. Potentials on Special Open Sets

      • J. L. Doob
      Pages 45-56

    6. Polar Sets and Their Applications

      • J. L. Doob
      Pages 57-69

    7. The Fundamental Convergence Theorem and the Reduction Operation

      • J. L. Doob
      Pages 70-84

    8. Green Functions

      • J. L. Doob
      Pages 85-97

    9. The Dirichlet Problem for Relative Harmonic Functions

      • J. L. Doob
      Pages 98-140

    10. Lattices and Related Classes of Functions

      • J. L. Doob
      Pages 141-154

    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

    17. Subparabolic, Superparabolic, and Parabolic Functions on a Slab

      • J. L. Doob
      Pages 285-294

    18. Parabolic Potential Theory (Continued)

      • J. L. Doob
      Pages 295-328

    19. The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets

      • J. L. Doob
      Pages 329-362
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Keywords

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.

Authors and Affiliations

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

    J. L. Doob

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