Article

Journal of Theoretical Probability

, Volume 23, Issue 4, pp 972-1001

Regularity of Intersection Local Times of Fractional Brownian Motions

  • Dongsheng WuAffiliated withDepartment of Mathematical Sciences, University of Alabama in Huntsville Email author 
  • , Yimin XiaoAffiliated withDepartment of Statistics and Probability, Michigan State University

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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.

Keywords

Intersection local time Fractional Brownian motion Joint continuity Hölder condition Hausdorff dimension Packing dimension

Mathematics Subject Classification (2000)

60G15 60J55 60G18 60F25 28A80