Skip to main content
SpringerLink
Log in

Hölder properties of local times for fractional Brownian motions

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

We study the local times of fractional Brownian motions for all temporal dimensions, N, spatial dimensions d and Hurst parameters H for which local times exist. We establish a Hölder continuity result that is a refinement of Xiao (Probab Th Rel Fields 109:129–157, 1997). Our approach is an adaptation of the more general attack of Xiao (Probab Th Rel Fields 109:129–157, 1997) using ideas of Baraka and Mountford (1997, to appear), the principal result of this latter paper is contained in this article.

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

  • Adler RJ (1981) The geometry of random fields. Wiley, New York

    MATH  Google Scholar 

  • Baraka D, Mountford T (1997) A law of iterated logarithm for fractional Brownian motions (to appear)

  • Baraka D, Mountford T (2008) The exact Hausdorff measure of the zero set of fractional Brownian motion (to appear)

  • Embrechts P, Maejima M (2002) Selfsimilar processes. Princeton University Press, Princeton

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kasahara Y (1978) Tauberian theorems of exponential type. J Math Kyoto Univ 18: 209–219

    MATH  MathSciNet  Google Scholar 

  • Kasahara Y, Kôno N, Ogawa T (1999) On tail probability of local times of Gaussian processes. Stoch Process Appl 82: 15–21

    Article  MATH  Google Scholar 

  • Mountford T (2007) Level sets of multiparameter stable processes. J Theoret Probab 20: 25–46

    Article  MATH  MathSciNet  Google Scholar 

  • Mountford T, Nualart E (2004) Level sets of multiparameter Brownian motions. Electron J Probab 9: 594–614

    MathSciNet  Google Scholar 

  • Mountford T, Shieh NR, Xiao Y (2008) The tail probabilities for local times of fractional Brownian motion (to appear)

  • Pitt LD, Tran LT (1979) Local sample path properties of Gaussian fields. Ann Probab 7: 477–493

    Article  MATH  MathSciNet  Google Scholar 

  • Ross S (1996) Initiation aux Probabilités. Presses Polytechniques et Universitaires Romandes, Lausanne

    Google Scholar 

  • Rogers CA (1998) Hausdorff measures. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian random processes: stochastic models with infinite variance, stochastic modeling. Chapman & Hall, New York

    Google Scholar 

  • Talagrand M (1995) Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann Probab 23: 767–775

    Article  MATH  MathSciNet  Google Scholar 

  • Talagrand M (1998) Multiple points of trajectories of multiparameter fractional Brownian motion. Probab Th Rel Fields 112: 545–563

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Xiao Y (2006) Properties of local-nondeterminism of Gaussian and stable random fields and their applications. Ann Fac Sci Toulouse Math (6) 15: 157–193

    MATH  MathSciNet  Google Scholar 

  • Xiao Y (2007) 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. Higher Education Press, Beijing, pp 136–176

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques, École Polytechnique Fédéralede Lausanne, Station 8, 1015, Lausanne, Switzerland

    D. Baraka & T. Mountford

  2. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Y. Xiao

Authors
  1. D. Baraka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Mountford
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Y. Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Baraka.

Additional information

Research of Y. Xiao is partially supported by NSF grant DMS-0706728.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baraka, D., Mountford, T. & Xiao, Y. Hölder properties of local times for fractional Brownian motions. Metrika 69, 125–152 (2009). https://doi.org/10.1007/s00184-008-0211-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-008-0211-6

Keywords

Search

Navigation