Intersection local times of independent fractional Brownian motions as generalized white noise functionals

TL;DR

This paper develops a generalized white noise functional expansion for intersection local times of independent fractional Brownian motions across any dimension and Hurst parameters, providing new conditions for their existence.

Contribution

It introduces a novel expansion of intersection local times in terms of Wick powers of white noises for fractional Brownian motions with arbitrary Hurst coefficients.

Findings

Derived a sufficient condition on dimension for L^2 existence of intersection local times.

Extended previous results to more general Hurst coefficients.

Provided a framework for analyzing intersection local times in generalized white noise settings.

Abstract

In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and more general Hurst coefficients.