Skip to main content

In Search of Multiparameter Markov Processes

  • Chapter
  • 158 Accesses

Abstract

One of the problems that John Thomas and I first tackled concerned the transition density of Markov processes. Through the Fokker-Planck equation we were able to show that the polynomial expansions first noted by Barrett and Lampard [1] could be interpreted in terms of the Sturm-Liouville problem to which the Fokker-Planck equation gives rise. My interest in Markov processess, kindled then, has never waned.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • 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.

References

  1. J.F. Barrett and D.G Lampard, An expansion for some second-order probability distributions and its applications to noise problems, IRE Trans, on Inform. Theory, Vol. IT-1, pp. 10–15, 1955.

    Google Scholar 

  2. P. Lévy, A special problem of Brownian motion, and a general theory of Gaussian random functions, Proc. 3rd Berkeley Symp. Math. Stat, and Prob., Vol. 2, Berkeley, CA: Univ. of California Press, pp. 133–175. 1956.

    Google Scholar 

  3. H.P. McKean Jr., Brownian motion with a second dimensional term, Theory Prob. Appl., Vol. 8, pp. 335–365, 1963.

    Article  MathSciNet  Google Scholar 

  4. R. Gangoli, Abstract harmonic analysis and Lévy’s Brownian motion of several parameters, Proc. 5th Berkeley Symp. Math. Stat, and Prob., Vol. II-l, Berkeley, CA: Univ. of California Press, Vol. II–l, pp. 13–30, 1967.

    Google Scholar 

  5. E. Wong, Homogeneous Gauss-Markov random fields, Ann. Math. Statist., Vol. 40, pp. 1625–1634, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Nelson, The free Markov field, J. Functional Analysis, Vol. 12, pp. 211–227, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Wong and M. Zakai, Martingales and stochastic integrals for processes with a multidimensional parameter, Z. Wahr. verw. Geb., Vol. 29, pp. 109–122, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Cairoli and J. Walsh, Stochastic integrals in the plane, Acta Math. Vol. 134, pp. 111–183, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Walsh, Martingales with a multidimensional parameter and stochastic integrals in the plane, Notes for cours de troisième cycle, University of Paris, V I, 1977.

    Google Scholar 

  10. C. Belisle, Private communication, 1983.

    Google Scholar 

  11. K. Itô Isotropic random current, Proc. 3rd Berkeley Symp. Math. Stat, and Prob., Vol. 2, Berkeley, CA: Univ. of California Press, pp. 125–132, 1956.

    Google Scholar 

  12. 12]H. Whitney, Geometric Integration Theory, Princeton, NJ: Princeton University Press, 1957.

    Google Scholar 

  13. B. Hajek and E. Wong, Set parametered martingales and multiple stochastic integration, Lecture Notes in Mathematics, No. 851, Berlin: Springer-Verlag, pp. 119–151, 1980.

    Google Scholar 

  14. E. Wong and M. Zakai, Differentiation formulas for stochastic integrals in the plane, Stock Proc. Appl., Vol. 6, pp. 339–349, 1978.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

Wong, E. (1986). In Search of Multiparameter Markov Processes. In: Blake, I.F., Poor, H.V. (eds) Communications and Networks. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4904-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-4904-7_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-9354-5

  • Online ISBN: 978-1-4612-4904-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics