Skip to main content
SpringerLink
Log in
Book cover
  • Book
  • © 1984

Classical Potential Theory and Its Probabilistic Counterpart

Advanced Problems

Authors:

  1. J. L. Doob

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

    View author publications

    You can also search for this author in PubMed   Google Scholar

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

Sections

Buy it now

eBook USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Other ways to access

This is a preview of subscription content, log in via an institution to check for access.

Table of contents (32 chapters)

  1. Classical and Parabolic Potential Theory

  2. Probabilistic Counterpart of Part 1

    1. Front Matter

      Pages 385-385
      PDF

    2. Fundamental Concepts of Probability

      • J. L. Doob
      Pages 387-412

    3. Optional Times and Associated Concepts

      • J. L. Doob
      Pages 413-431

    4. Elements of Martingale Theory

      • J. L. Doob
      Pages 432-462

    7. Markov Processes

      • J. L. Doob
      Pages 539-569

    8. Brownian Motion

      • J. L. Doob
      Pages 570-598

    9. The Itô Integral

      • J. L. Doob
      Pages 599-626

    10. Brownian Motion and Martingale Theory

      • J. L. Doob
      Pages 627-667

    11. Conditional Brownian Motion

      • J. L. Doob
      Pages 668-702

  3. Part 3

    1. Front Matter

      Pages 703-703
      PDF

    3. Brownian Motion and the PWB Method

      • J. L. Doob
      Pages 719-726

    4. Brownian Motion on the Martin Space

      • J. L. Doob
      Pages 727-738

  4. Back Matter

    Pages 739-847
    PDF
Back to top

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.
Back to top

Keywords

Back to top

Authors and Affiliations

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

    J. L. Doob

Back to top

Bibliographic Information

Back to top

Publish with us

Policies and ethics

Back to top
Access via your institution

Buy it now

eBook USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Other ways to access

Search

Navigation