Journal of Theoretical Probability

, Volume 28, Issue 1, pp 299–312 | Cite as

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

  • Paul Jung
  • Greg Markowsky


We prove joint Hölder continuity and an occupation-time formula for the self-intersection local time of fractional Brownian motion. Motivated by an occupation-time formula, we also introduce a new version of the derivative of self-intersection local time for fractional Brownian motion and prove Hölder conditions for this process. This process is related to a different version of the derivative of self-intersection local time studied by the authors in a previous work.


Intersection local time Fractional Brownian motion  Occupation-time formula 

Mathematics Subject Classification (2010)

60G22 60J55 



We thank Yimin Xiao for helpful comments. The second author was supported by Australian Research Council Grant DP0988483.


  1. 1.
    Chen X.: Random walk intersections: large deviations and related topics, vol 157. Amer Mathematical Society (2010)Google Scholar
  2. 2.
    Dynkin E.B.: Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 1–57 (1988)Google Scholar
  3. 3.
    Garsia A.M.: Topics in almost everywhere, convergence. Markham (1970)Google Scholar
  4. 4.
    Berman, S.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23(1), 69–94 (1974)CrossRefGoogle Scholar
  5. 5.
    Hu, Y.: Self-intersection local time of fractional Brownian motions–via chaos expansion. J. Math. Kyoto Univ. 41(2), 233–250 (2001)MATHMathSciNetGoogle Scholar
  6. 6.
    Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33(3), 948–983 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hu, Y., Nualart, D.: Central limit theorem for the third moment in space of the Brownian local time increments. Electron. Commun. Probab. 15, 396–410 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Jung P., Markowsky G.: On the tanaka formula for the derivative of self-intersection local time of fbm. Arxiv, preprint arXiv:1205.5551 (2012)Google Scholar
  9. 9.
    Markowsky G.: Proof of a Tanaka-like formula stated. In: Rosen J. (ed.) Séminaire XXXVIII. Séminaire de Probabilités XLI, pp. 199–202 (2008)Google Scholar
  10. 10.
    Markowsky, G.: Renormalization and convergence in law for the derivative of intersection local time in \(\mathbb{R}^2\). Stoch. Process. Appl. 118(9), 1552–1585 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer Verlag, New York (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Rogers, L.C.G., Walsh, J.B.: Local time and stochastic area integrals. Ann. Probab. 19(2), 457–482 (1991)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Rosen, J.: The intersection local time of fractional Brownian motion in the plane. J. Multivar. Anal. 23(1), 37–46 (1987)CrossRefMATHGoogle Scholar
  14. 14.
    Rosen, J.: Limit laws for the intersection local time of stable processes in \(R^2\). Stochastics 23(2), 219–240 (1988)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Rosen J.: Derivatives of self-intersection local times. Séminaire de Probabilités XXXVIII, pp. 263–281 (2005)Google Scholar
  16. 16.
    Varadhan, S.R.S.: Appendix to Euclidian quantum field theory. In: Symanzy, K. (ed.) Local Quantum Theory. Academic Press, New York (1969)Google Scholar
  17. 17.
    Wu, D., Xiao, Y.: Regularity of intersection local times of fractional brownian motions. J. Theor. Probab. 23(4), 972–1001 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Xiao, Y.: Holder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Relat. Fields 109(1), 129–157 (1997)CrossRefMATHGoogle Scholar
  19. 19.
    Yan, L., Yang, X., Lu, Y.: p-Variation of an integral functional driven by fractional Brownian motion. Stat. Probab. Lett. 78(9), 1148–1157 (2008)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Yan, L., Liu, J., Yang, X.: Integration with respect to fractional local time with Hurst index \(1/2<H<1\). Potential Anal. 30(2), 115–138 (2009)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlabamaBirminghamUSA
  2. 2.Department of MathematicsMonash UniversityClaytonAustralia

Personalised recommendations