Regularity of Intersection Local Times of Fractional Brownian Motions

TL;DR

This paper investigates the existence, regularity, and geometric properties of intersection local times of independent fractional Brownian motions with different Hurst indices, revealing how anisotropy influences these properties.

Contribution

It extends previous work by analyzing intersection local times for anisotropic fractional Brownian motions with different Hurst indices, providing new regularity and geometric results.

Findings

Existence of intersection local times under certain conditions

Hölder continuity and regularity properties established

Determination of Hausdorff and packing dimensions of intersection sets

Abstract

Let $B^{\alpha_i}$ be an $(N_i,d)$-fractional Brownian motion with Hurst index ${\alpha_i}$ ($i=1,2$), 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 H\"{o}lder 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 ({\it J. Theor. Probab.} {\bf 20} (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.