Intersection local times of fractional Brownian motions with $H\in(0,1)$ as generalized white noise functionals

TL;DR

This paper develops expansions for the self-intersection local times of fractional Brownian motions across all dimensions and Hurst parameters, using generalized white noise functionals and Wick powers.

Contribution

It introduces a novel expansion framework for fractional Brownian motion local times in terms of Wick powers, applicable to all dimensions and Hurst coefficients.

Findings

Explicit expansions in terms of Wick powers for all Hurst parameters

Well-defined in the sense of generalized white noise functionals

Applicable to any dimension $d \\geq 1$

Abstract

In $\R^d$, for any dimension $d\geq 1$, expansions of self-intersection local times of fractional Brownian motions with arbitrary Hurst coefficients in $(0,1)$ are presented. 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.