Journal of Theoretical Probability

, Volume 11, Issue 2, pp 303–330

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

  • René L. Schilling


Let {Xt}t≥0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇ∈∝n|q(Ç,ξ)|≤c(1=ψ)(ξ)) for some fixed continuous negative definite function ψ(ξ). The Hausdorff dimension of the set {Xt:t∈E}, E ⊂ [0, 1] is any analytic set, is a.s. bounded above by βψ dim E. βψ is the Blumenthal−Getoor upper index of the Levy Process associated with ψ(ξ).

Levy-type process Hausdorff dimension Feller semigroup pseudo-differential operator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berg, C., and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups. Springer, Ergebnisse der Mathematik and ihrer Grenzgebiete, II. Ser. Vol. 87, Berlin.Google Scholar
  2. 2.
    Blumenthal, R. M., and Getoor, R. K. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10, 493–516.Google Scholar
  3. 3.
    Bochner, S. (1955). Harmonic Analysis and the Theory of Probability. University of California Press, California Monographs in Math. Sci., Berkeley, California.Google Scholar
  4. 4.
    Courrège, Ph. (1966). Sur la forme intégro-différentielle des opérateurs de C k and C satisfaisant au principe du maximum. Sém. Théorie du Potentiel, p. 38.Google Scholar
  5. 5.
    Ethier, St. E., and Kurtz, Th. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, Series in Prob. and Math. Stat., New York.Google Scholar
  6. 6.
    Fristedt, B. (1974). Sample functions of stochastic processes with stationary, independent increments. In Ney, P., and Port, S., (eds.), Advances in Probability and related Topics, Vol. 3, Dekker, New York, pp. 241–396.Google Scholar
  7. 7.
    Gradshteyn, I., Ryzhik, I., and Jeffrey, A., (eds.), (1993). Tables of Integrals, Series, and Products. Fifth Edition, Academic Press, Boston, Massachusetts.Google Scholar
  8. 8.
    Hawkes, J. (1971). Some dimension theorems for the sample functions of stable processes. Indiana J. Math. 20, 733–738.Google Scholar
  9. 9.
    Hawkes, J., and Pruitt, W. E. (1974). Uniform dimension results for processes with independent increments. Z. Wahrsch. verw. Geb. 28, 277–288.Google Scholar
  10. 10.
    Hoh, W. (1993). Some commutator estimates for pseudo-differential operators with negative definite functions as symbols. Integr. Equat. Oper. Th. 17, 46–53.Google Scholar
  11. 11.
    Hoh, W. (1995). Pseudo-differential operators with negative definite symbols and the martingale problem. Stoch. and Stoch. Rep. 55, 225–252.Google Scholar
  12. 12.
    Jacob, N. (1994). A class of Feller semigroups generated by pseudo-differential operators. Math. Z. 215, 151–166.Google Scholar
  13. 13.
    Jacob, N. (1995). Characteristic functions and symbols in the theory of Feller processes. Potential Analysis (to appear).Google Scholar
  14. 14.
    Jacob, N. (1996). Pseudo-differential operators and Markov processes. Akademie-Verlag, Mathematical Research, Vol. 94, Berlin.Google Scholar
  15. 15.
    Jacob, N., and Schilling, R. L. (1996). Estimates for Feller semigroups generated by pseudodifferential operators. In Rákosník, J., (ed.), Function Spaces, Differential Operators and Nonlinear Analysis Proc. Inter. Conf., Paseky nad Jizerou, September 3–9, 1995, Prometheus Publishing House, 27–49.Google Scholar
  16. 16.
    Lepeltier, J.-P., and Marchal, B. (1976). Problème de martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel. Ann. Inst. Henri Poincaré B 12, 43–103.Google Scholar
  17. 17.
    Millar, P. W. (1971). Path behavior of Processes with stationary independent increments. Z. Wahrsch. verw. Geb. 17, 53–73.Google Scholar
  18. 18.
    Negoro, A. (1994). Stable-like processes: construction of the transition density and the behavior of sample paths near t = 0. Osaka J. Math. 31, 189–214.Google Scholar
  19. 19.
    Pruitt, W. E. (1969). The Hausdorff dimension of the range of a process with stationary independent increments. Indiana J. Math. 19, 371–378.Google Scholar
  20. 20.
    Schilling, R. L. (1994). Zum Pfadverhalten von Markovschen Prozessen, die mit Lévy-Prozessen vergleichbar sind, Dissertation, Universität Erlangen.Google Scholar
  21. 21.
    Schilling, R. L. (1996). Comparable processes and the Hausdorff dimension of their sample paths. Stoch. and Stoch. Rep. 57, 89–110.Google Scholar
  22. 22.
    Schilling, R. L. (undated). Conversativeness of semigroups generated by pseudo-differential operators. Potential Analysis (to appear).Google Scholar
  23. 23.
    Taylor, S. J. (1973). Sample path properties of processes with stationary independent increments. In Kendall, D. G., and Harding, E. F., (eds.), Stochastic Analysis. John Wiley, London, 387–414.Google Scholar
  24. 24.
    Taylor, S. J. (1986). The measure theory of random fractals, Math. Proc. Camb. Phil. Soc. 100, 383–406.Google Scholar
  25. 25.
    Tsuchyia, M. (1992). Lévy measure with generalized polar decomposition and the associated SDE with jumps. Stoch. and Stoch. Rep. 38, 95–117.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • René L. Schilling

There are no affiliations available

Personalised recommendations