Skip to main content

Potential theory for recurrent symmetric infinitely divisible processes

  • 461 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 928)

Abstract

Let X be a locally compact, second countable Abelian group. Let ϕ(t), t≥0, be an irreducible, recurrent, symmetric infinitely divisible Hunt process on X such that for t>0, ϕ(t)-ϕ(0) has a bounded continuous density p(t,·) with respect to Haar measure on X. Potential theory is developed for the kernel k = ∫ 10 p(t,·)dt · ∫ 1 (p(t,·)−p(t,0))dt. In particular, balayage and equilibrium problems corresponding to an arbitrary relatively compact Borel set are formulated and solved and the solutions are characterized in terms of energy. Logarithmic potential theory is included as the special case corresponding to planar Brownian motion.

Keywords

  • Compact Subset
  • Equilibrium Problem
  • Lower Semicontinuous
  • Haar Measure
  • Finite Energy

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. R.M. BLUMENTHAL and R.K. GETOOR, Markov Processes and Potential Theory, Academic Press, New York, 1968.

    MATH  Google Scholar 

  2. E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis I, Springer-Verlag, Berlin, 1963.

    CrossRef  MATH  Google Scholar 

  3. S. C. PORT and C. J. STONE, Potential theory of random walks on Abelian groups, Acta Math., 122 (1969), 19–114.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. S. C. PORT and C. J. STONE, Infinitely divisible processes and their potential theory, Ann. Inst. Fourier 21(1971) (2) 157–275 and (4) 179–265.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. S. C. PORT and C. J. STONE, Brownian Motion and Classical Potential Theory, Academic Press, New York, 1978.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1982 Springer-Verlag

About this paper

Cite this paper

Stone, C.J. (1982). Potential theory for recurrent symmetric infinitely divisible processes. In: Heyer, H. (eds) Probability Measures on Groups. Lecture Notes in Mathematics, vol 928. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093235

Download citation

  • DOI: https://doi.org/10.1007/BFb0093235

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11501-4

  • Online ISBN: 978-3-540-39206-4

  • eBook Packages: Springer Book Archive