Fine properties of fractional Brownian motions on Wiener space

TL;DR

This paper investigates detailed path properties of fractional Brownian motions on Wiener space, including regularity, continuity, and law of iterated logarithm, using advanced Malliavin calculus techniques.

Contribution

It introduces a comprehensive analysis of fractional Brownian motions' fine properties on Wiener space, extending existing methods to this class of processes.

Findings

Proves non-differentiability of fractional Brownian paths

Establishes modulus of continuity results

Derives law of iterated logarithm for fractional Brownian motion

Abstract

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter $H$ under the $(p,r)$-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener functionals via the integral representation discovered by Decreusefond and \"{U}st\"{u}nel, and show non differentiability, modulus of continuity, law of iterated Logarithm(LIL) and self-avoiding properties of fractional Brownian motion sample paths using Malliavin calculus as well as the tools developed in the previous work by Fukushima, Takeda and etc. for Brownian motion case.