Skip to main content
SpringerLink
Log in

Regularity of Intersection Local Times of Fractional Brownian Motions

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Let \(B^{\alpha_{i}}\) be an (N i ,d)-fractional Brownian motion with Hurst index α i (i=1,2), and let \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) be independent. We prove that, if \(\frac{N_{1}}{\alpha_{1}}+\frac{N_{2}}{\alpha_{2}}>d\) , then the intersection local times of \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points.

One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre (J. Theor. Probab. 20:759–767, 2007), where the existence of the intersection local times of two independent (1,d)-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, R.J.: The Geometry of Random Fields. Wiley, New York (1981)

    MATH  Google Scholar 

  2. Ayache, A., Wu, D., Xiao, Y.: Joint continuity of the local times of fractional Brownian sheets. Ann. Inst. H. Poincaré Probab. Stat. 44, 727–748 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baraka, D., Mountford, T.: A law of the iterated logarithm for fractional Brownian motions. In: Séminaire de Probabilités XLI. Lecture Notes in Math., vol. 1934, pp. 161–179. Springer, Berlin (2008)

    Chapter  Google Scholar 

  4. Cuzick, J., DuPreez, J.: Joint continuity of Gaussian local times. Ann. Probab. 10, 810–817 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Falconer, K.J.: Fractal Geometry—Mathematical Foundations and Applications. Wiley, Chichester (1990)

    MATH  Google Scholar 

  6. Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8, 1–67 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Geman, D., Horowitz, J., Rosen, J.: A local time analysis of intersections of Brownian paths in the plane. Ann. Probab. 12, 86–107 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33, 948–983 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields. Springer, New York (2002)

    MATH  Google Scholar 

  10. LeGall, J.F.: Sur le temps local d’intersection du mouvement Brownien plan et la méthode de renormalisation de Varadhan. In: Séminaire de Probabilités XIX. Lecture Notes in Math., vol. 1123, pp. 314–331. Springer, Berlin (1985)

    Google Scholar 

  11. Monrad, D., Pitt, L.D.: Local nondeterminism and Hausdorff dimension. In: Cinlar, E., Chung, K.L., Getoor, R.K. (eds.) Progress in Probability and Statistics, pp. 163–189. Seminar on Stochastic Processes, 1986. Birkhauser, Boston (1987)

    Google Scholar 

  12. Nualart, D., Ortiz-Latorre, S.: Intersection local time for two independent fractional Brownian motions. J. Theor. Probab. 20, 759–767 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309–330 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pitt, L.D., Tran, L.T.: Local sample path properties of Gaussian fields. Ann. Probab. 7, 477–493 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Rogers, C.A., Taylor, S.J.: Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8, 1–31 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  16. Rosen, J.: The intersection local time of fractional Brownian motion in the plane. J. Multivar. Anal. 23, 37–46 (1987)

    Article  MATH  Google Scholar 

  17. Wolpert, R.: Wiener path intersections and local time. J. Funct. Anal. 30, 329–340 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wolpert, R.: Local time and a practice picture for Euclidean field theory. J. Funct. Anal. 30, 341–357 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  19. Xiao, Y.: Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Relat. Fields 109, 129–157 (1997)

    Article  MATH  Google Scholar 

  20. Xiao, Y.: Hitting probabilities and polar sets for fractional Brownian motion. Stoch. Stoch. Rep. 66, 121–151 (1999)

    MATH  Google Scholar 

  21. Xiao, Y.: Properties of local nondeterminism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV, 157–193 (2006)

    Google Scholar 

  22. Xiao, Y.: Strong local nondeterminism of Gaussian random fields and its applications. In: Lai, T.-L., Shao, Q.-M., Qian, L. (eds.) Asymptotic Theory in Probability and Statistics with Applications, pp. 136–176. Higher Education, Beijing (2007)

    Google Scholar 

  23. Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math., vol. 1962, pp. 145–212. Springer, New York (2009)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Alabama in Huntsville, 201J Shelby Center, Huntsville, AL, 35899, USA

    Dongsheng Wu

  2. Department of Statistics and Probability, Michigan State University, A-413 Wells Hall, East Lansing, MI, 48824, USA

    Yimin Xiao

Authors
  1. Dongsheng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yimin Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongsheng Wu.

Additional information

Y. Xiao’s research partially supported by NSF grant DMS-0706728.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, D., Xiao, Y. Regularity of Intersection Local Times of Fractional Brownian Motions. J Theor Probab 23, 972–1001 (2010). https://doi.org/10.1007/s10959-009-0221-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-009-0221-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation