Skip to main content
SpringerLink
Log in

Transience, Recurrence and Invariant Measures for Diffusions

  • Chapter
Nonlinear Stochastic Problems

Part of the book series: NATO ASI Series ((ASIC,volume 104))

Abstract

Qualitative theory describes the long term behavior of stochastic systems without solving the equations. In this paper we concentrate on transience, recurrence and ergodic properties of Markov solutions of stochastic differential equations. For several approaches the relations between the regularity of the transition semigroup (strong Markov — Feller — strong Feller-nondegeneracy) and the adequat topology are discussed. Results are given for diffusions, i.e. Feller processes with continuous trajectories on some state space X ⊂ ℝd, endowed with the usual topology from ℝd, because this is the situation amenable for applications.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold, L.; Kliemann, W. Qualitative theory of stochastic systems, to appear in: Probabilistic analysis and related topics, Vol. 3, Bharucha-Reid, A. T. (ed.), New York: Academic Press, 1982

    Google Scholar 

  2. Arnold, L.; Lefever, R. (eds.) Stochastic nonlinear systems. Berlin: Springer Verlag, 1981

    MATH  Google Scholar 

  3. Azéma, J.; Kaplan-Duflo, M.; Revuz, D. Récurrence fine des processus de Markov. Ann. Inst. Henri Poincaré, 2 (1966) p. 185–220

    MATH  Google Scholar 

  4. Mesure invariante sur les classes recurrentes des processus de Markov. Z. Wtheorie verw. Gebiete 8 (1967) p. 157–181

    Google Scholar 

  5. Note sur la mesure invariante des processus de Markov recurrents. Ann. Inst. Henri Poincaré, 3 (1967) p. 397–402

    Google Scholar 

  6. Propriétés relatives des processus de Markov recurrents. Z. Wtheorie verw. Gebiete 13 (1969) p. 286–314

    Google Scholar 

  7. Bhatia, N. P.; Szegö, G. P. Stability theory of dynamical systems. Berlin: Springer Verlag, 1970

    MATH  Google Scholar 

  8. Bhattacharya, R. N. Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Prob. 6 (1978) p. 541–553 and 8 (1980) p. 1194–1195

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Duflo, M.; Revuz, D. Propriétés asymptotiques des probabilités de transition des processus de Markov recurrents, Ann. Inst. Henri Poincare, 5 (1969) p. 233–244

    MathSciNet  MATH  Google Scholar 

  11. Dynkin, E. B. Markov processes, I and II, Berlin: Springer Verlag, 1965

    Google Scholar 

  12. Friedman, A. Wandering out to infinity of diffusion processes. Trans. AMS 184 (1973) p. 185–203

    Google Scholar 

  13. Stochastic differential equations and applications, I and II. New York: Academic Press, 1975 and 1976

    Google Scholar 

  14. Getoor, R. K. Transience and recurrence of Markov processes. Lecture Notes in Mathematics 784, Seminaire de Probability XIV (1980) p. 397–409.

    Article  MathSciNet  Google Scholar 

  15. Hasminskii (Khasminkii), R. Z. Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theor. Prob. Appl. 5 (1960) p. 179–195

    Article  Google Scholar 

  16. Stochastic stability of differential equations. Alphen aan den Rijn: Sijthoff and Noordhoff, 1980, (russian: 1969 )

    Google Scholar 

  17. Ichihara, K., Kunita, H. A. Classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wtheorie verw. Gebiete 30(1974) p. 235–254 and 39(1977) p. 81-84

    Article  MathSciNet  MATH  Google Scholar 

  18. Ikeda, N.; Watanabe, S. Stochastic differential equations and diffusion processes. Amsterdam: North Holland. 1981

    MATH  Google Scholar 

  19. Ito, K. On stochastic differential equations. Mem. Amer. Math. Soc. 4; 1951

    Google Scholar 

  20. Kliemann, W. Qualitative theory of stochastic dynamical systems applications to life sciences, to appear in: Bull. Math. Bio. (1982)

    Google Scholar 

  21. Diffusions on manifolds. Report “Forschungs-schwerpunkt Dynamische Systeme” University of Bremen, 1982

    Google Scholar 

  22. Kushner, H. Stochastic stability and control. New York: Academic Press, 1967

    MATH  Google Scholar 

  23. Miyahara, Y. Ultimate boundedness of the systems governed by stochastic differential equations. Nagoya Math. J. 47(1972) p. 111–144.

    MathSciNet  MATH  Google Scholar 

  24. Invariant measures of ultimately bounded stochastic processes. Nagoya Math. J. 49 (1973) p. 149–153

    Google Scholar 

  25. Nemytskii, V. V.; Stepanov, V. V. Qualitative theory of differential equations. Princeton, 1960.

    MATH  Google Scholar 

  26. Stroock, D. W.; Varadhan, S. R. S. On the support of diffusion processes with applications to the strong maximum principle. Proc. Sixth Berkeley Symp. Vol. III (1972) p. 333–359.

    MathSciNet  Google Scholar 

  27. On degenerate elliptic parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math. 25 (1972) p. 651–713

    Google Scholar 

  28. Multidimensional diffusion processes. Berlin: Springer Verlag, 1979

    Google Scholar 

  29. Williams, D. Diffusions, Markov processes and martingales. New York: Wiley, 1979

    MATH  Google Scholar 

  30. Wonham, W. M. Lyapunov criteria for weak stochastic stability. J. Diff. Equ. 2 (1966) p. 365–377.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Forschungsschwerpunkt “Dynamische Systeme”, University of Bremen, 2800, Bremen, West Germany

    W. Kliemann

Authors
  1. W. Kliemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Aerospace Engineering, University of Southern California, Los Angeles, CA, USA

    Richard S. Bucy

  2. Department of Electrical Engineering, Instituto Superior Técnico, Lisbon, Portugal

    José M. F. Moura

Rights and permissions

Reprints and permissions

Copyright information

© 1983 D. Reidel Publishing Company

About this chapter

Cite this chapter

Kliemann, W. (1983). Transience, Recurrence and Invariant Measures for Diffusions. In: Bucy, R.S., Moura, J.M.F. (eds) Nonlinear Stochastic Problems. NATO ASI Series, vol 104. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7142-4_32

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-7142-4_32

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-7144-8

  • Online ISBN: 978-94-009-7142-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation