Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Capacity theory without duality
Download PDF
Download PDF
  • Published: September 1986

Capacity theory without duality

  • R. K. Getoor1 &
  • J. Steffens2 

Probability Theory and Related Fields volume 73, pages 415–445 (1986)Cite this article

  • 70 Accesses

  • 8 Citations

  • Metrics details

Summary

The paper develops a theory of capacity for a Borel right process without duality assumptions. The basic tool in this approach is a stationary process ralative to an excessive measure.

IfP t )t≧0 denotes the semigroup of the process on the state spaceE and ifm is an excessive measure onE, then there exists a processY = (Y t ) t∈ℝ onE with random birth and death and a δ-finite measureQ m such thatY is stationary underQ m and Markov with respect to (P t ).

For a setB inE the hitting (resp. last exit) time ofY is denoted by τ B (resp.λ B ), andB is called transient (resp. cotransient) ifQ m (λ B =∞)= 0 (resp.Q m (τ B = − ∞)=0. The main theorem then states that for a both transient and contransient setB the distributions ofλ B and τ B underQ m are the same. For suchB the capacity is denfined byC(B):=Q m (λ B ∈[0, 1] and the cocapacity by\(\hat C\)(B):=Q m (τ B ∈[0, 1], and it is shown that these definitions in fact generalize previous definitions under duality assumptions.

Without duality assumption there is no representation of the capacitary potential in terms of a capacitary measure, but there exists a cocapacitary entrance law ρ B t which generalizes the notion of a cocapacitary measure such that\(\hat C\)(B)=\(\mathop {\sup }\limits_{t > 0} \) ρ B t (1).

The paper contains investigations of transience and cotransience, a decomposition of the cocapacitrary entrance law, some remarks on left versions, and furthermore a generalization of Spitzer's asymptotic formula.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Blumenthal, R.M.: A decomposition of excessive measures. Sem. Stoch. Proc. 1985. Boston: Birkhäuser. (To appear)

  2. Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968

    Google Scholar 

  3. Chung, K.L.: Probability approach to the equilibrium problem in potential theory. Ann. Inst. Fourier (Grenoble)23, 313–322 (1973)

    Google Scholar 

  4. Dellacherie, C., Meyer, P.A.: Probabilities and potential, vol. I. Amsterdam-New York-Oxford: North Holland 1978

    Google Scholar 

  5. Dynkin, E.B.: A new approach to Markov processes. Proc. Japan-USSR Symp. Prob. Theory. Lecture Notes in Math.550, 42–62. Berlin-Heidelberg-New York: Springer 1976

    Google Scholar 

  6. Dynkin, E.B.: Regular Markov processes. Russian Math. Surv.28, 33–64 (1973). Reprinted in London Math. Soc. Lecture Notes Series54. Cambridge: Cambridge Univ. Press 1982

    Google Scholar 

  7. Dynkin, E.B.: Minimal excessive functions and measures. Trans. Am. Math. Soc.258, 217–244 (1980)

    Google Scholar 

  8. Fitzsimmons, P.J., Maisonneuve, B.: Excessive measures and Markov processes with random birth and death. Z. Wahrscheinlichkeitstheor. Verw. Geb.72, 319–336 (1986)

    Google Scholar 

  9. Getoor, R.K.: Markov processes: Ray processes and right processes. Lecture Notes in Math.440. Berlin-Heidelberg-New York: Springer 1975

    Google Scholar 

  10. Getoor, R.K.: Some asymptotic formulas involving capacity. Z. Wahrscheinlichkeitstheor. Verw. Geb.4, 248–252 (1965)

    Google Scholar 

  11. Getoor, R.K.: Transience and recurrence of Markov processes. Sem. de Prob. XIV. Lecture Notes in Math.784, 397–409. Berlin-Heidelberg-New York: Springer 1980

    Google Scholar 

  12. Getoor, R.K.: Capacity theory and weak duality. Sem. Stoch. Proc. 1983, pp. 97–130. Boston: Birkhäuser 1984

    Google Scholar 

  13. Getoor, R.K.: Some remarks on measures associated with homogeneous random measures. Sem. Stoch. Proc. 1985. Boston: Birkhäuser (To appear)

  14. Getoor, R.K., Glover, J.: Markov processes with identical excessive measures. Math. Z.184, 287–300 (1983)

    Google Scholar 

  15. Getoor, R.K., Glover, J.: Riez decompositions in Markov process theory. Trans. Am. Math. Soc.285, 107–132 (1984)

    Google Scholar 

  16. Getoor, R.K., Sharpe, M.J.: Last exit times and additive functionals. Ann. Probab.1, 550–569 (1973)

    Google Scholar 

  17. Getoor, R.K., Sharpe, M.J.: Naturality, standardness, and weak duality for Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 1–62 (1984)

    Google Scholar 

  18. Hunt, G.A.: Markov processes and potentials I., III. Ill. J. Math.1, 44–93,2, 151–213 (1958)

    Google Scholar 

  19. Kuznetsov, S.E.: Construction of Markov processes with random times of birth and death. Theory Probab. Appl.18, 571–575 (1974)

    Google Scholar 

  20. Mitro, J.B.: Dual Markov processes: construction of a useful auxiliary process. Z. Wahrscheinlichkeitstheorie. Verw. Geb.47, 139–156 (1979)

    Google Scholar 

  21. Port, S.C., Stone, C.J.: Infinitely divisible processes and their potential theory I. Ann. Inst. Fourier (Grenoble)21, 157–275 (1971)

    Google Scholar 

  22. Spitzer, F.: Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 110–121 (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, 92093, La Jolla, CA, USA

    R. K. Getoor

  2. Institut für Statistik u. Dok., Universität Düsseldorf, Universitätsstrasse 1, D-4000, Düsseldorf 1, Germany

    J. Steffens

Authors
  1. R. K. Getoor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Steffens
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by NSF Grant DMS 8419377

Research carried out while visiting University of California, San Diego, during Spring 1985

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Getoor, R.K., Steffens, J. Capacity theory without duality. Probab. Th. Rel. Fields 73, 415–445 (1986). https://doi.org/10.1007/BF00776241

Download citation

  • Received: 23 August 1985

  • Revised: 06 March 1986

  • Issue Date: September 1986

  • DOI: https://doi.org/10.1007/BF00776241

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • State spaceE
  • Stochastic Process
  • Stationary Process
  • Probability Theory
  • Mathematical Biology
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature