Journal of Theoretical Probability

, Volume 23, Issue 4, pp 972–1001

Regularity of Intersection Local Times of Fractional Brownian Motions

Article

DOI: 10.1007/s10959-009-0221-y

Cite this article as:
Wu, D. & Xiao, Y. J Theor Probab (2010) 23: 972. doi:10.1007/s10959-009-0221-y

Abstract

Let \(B^{\alpha_{i}}\) be an (Ni,d)-fractional Brownian motion with Hurst index αi (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.

Keywords

Intersection local timeFractional Brownian motionJoint continuityHölder conditionHausdorff dimensionPacking dimension

Mathematics Subject Classification (2000)

60G1560J5560G1860F2528A80

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA