Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Markov functions of a time-changed recurrent diffusion

  • 44 Accesses

  • 1 Citations

Abstract

Let (X t ,P x ) be a recurrent diffusion on the state spaceE. A necessary and sufficient condition on the continuous functionu:ER is given so thatu is a Markov function for a time-changed diffusionX τ. It is shown that no nonconstant continuous real-valued function is Markov for a Brownian motion on the Sierpinski gasket.

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

References

  1. 1.

    Barlow, M. T., and Perkins, E. A. (1988). Brownian Motion on the Sierpinski Gasket,Prob. Th. Rel. Fields 79, 543–623.

  2. 2.

    Bliedtner, J., and Hansen, W. (1986).Potential Theory: An Analytic and Probabilistic Approach to Balayage, Springer, Berlin.

  3. 3.

    Blumenthal, R. G., and Getoor, R. K. (1968).Markov Processes and Potential Theory, Academic Press, New York.

  4. 4.

    Csink, L., Fitzsimmons, P. J., Øksendal, B. (1990). A Stochastic Characterization of Harmonic Morphisms.Math. Ann. 287(1), 1–18.

  5. 5.

    Dynkin, E. B. (1965).Markov Processes 1, Springer, Berlin.

  6. 6.

    Getoor, R. K. (1980). Transience and recurrence of Markov processes. In Azéma, J., Meyer, P. A., Yor, M. (eds).Séminaire de Probabilité XIV, Lecture Notes in Math. Springer, Berlin784, 397–409.

  7. 7.

    Glover, J. (1991). Markov Functions.Annales de l'Institute Henri Poincaré 27 221–238.

  8. 8.

    Glover, J. (1991). Application of Symmetry Groups in Markov Processes. In Heyer, H. (ed).Probability Measures on Groups X, Plenum Press, New York and London, pp. 155–168.

  9. 9.

    Glover, J. and Mitro, J. (1990). Symmetries and Functions of Markov Processes.Ann. Prob. 18, 655–668.

  10. 10.

    Harrison, M. J. (1985).Brownian Motion and Stochastic Flow Systems, John Wiley and Sons, New York.

  11. 11.

    Kigami, J. (1989). A Harmonic Calculus on the Sierpinski Spaces.Japan J. Appl. Math. 6(2), 259–290.

  12. 12.

    Revuz, D., and Yor, M. (1991).Continuous Martingales and Brownian Motion, Springer, Berlin.

  13. 13.

    Rogers, L. C. G., and Pitman, J. (1981). Markov functions.Ann. Prob. 9, 573–582.

  14. 14.

    Sharpe, M. (1989).General Theory of Markov Processes, Academic Press, New York.

  15. 15.

    Walsh, J. (1975). Functions of Brownian motion.Proc. Amer. Math. Soc. 49, 227–231.

  16. 16.

    Wang, A. T., and Chen, C. S. (1979). Functions ofn-dimensional Brownian motion that are Markovian.Israel J. Math. 34, 343–352.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Vondraček, Z. Markov functions of a time-changed recurrent diffusion. J Theor Probab 6, 485–497 (1993). https://doi.org/10.1007/BF01066714

Download citation

Key Words

  • Markov functions
  • diffusions
  • time-change
  • Brownian motion on the Sierpinski gasket