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.
Similar content being viewed by others
References
Adler RJ (1981) The geometry of random fields. Wiley, New York
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
Geman D, Horowitz J (1980) Occupation densities. Ann Probab 8: 1–67
Geman D, Horowitz J, Rosen J (1984) A local time analysis of intersections of Brownian paths in the plane. Ann Probab 12: 86–107
Kasahara Y (1978) Tauberian theorems of exponential type. J Math Kyoto Univ 18: 209–219
Kasahara Y, Kôno N, Ogawa T (1999) On tail probability of local times of Gaussian processes. Stoch Process Appl 82: 15–21
Mountford T (2007) Level sets of multiparameter stable processes. J Theoret Probab 20: 25–46
Mountford T, Nualart E (2004) Level sets of multiparameter Brownian motions. Electron J Probab 9: 594–614
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
Ross S (1996) Initiation aux Probabilités. Presses Polytechniques et Universitaires Romandes, Lausanne
Rogers CA (1998) Hausdorff measures. Cambridge University Press, Cambridge
Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian random processes: stochastic models with infinite variance, stochastic modeling. Chapman & Hall, New York
Talagrand M (1995) Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann Probab 23: 767–775
Talagrand M (1998) Multiple points of trajectories of multiparameter fractional Brownian motion. Probab Th Rel Fields 112: 545–563
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
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
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
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of Y. Xiao is partially supported by NSF grant DMS-0706728.
Rights 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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-008-0211-6