Sample Path Properties of the Fractional Wiener--Weierstrass Bridge

TL;DR

This paper studies the sample path properties of fractional Wiener--Weierstrass bridges, a class of Gaussian processes, by analyzing their continuity, variation, Hausdorff dimension, and maximum location using advanced fractional integral bounds.

Contribution

It introduces new bounds for fractional integrals and applies them to analyze the detailed sample path properties of fractional Wiener--Weierstrass bridges.

Findings

Established bounds for local and uniform moduli of continuity.

Determined the Hausdorff dimension of the sample paths.

Identified the location of the maximum of the processes.

Abstract

Fractional Wiener--Weierstrass bridges are a class of Gaussian processes that arise from replacing the trigonometric function in the construction of classical Weierstrass functions by a fractional Brownian bridge. We investigate the sample path properties of such processes, including local and uniform moduli of continuity, $\Phi$-variation, Hausdorff dimension, and location of the maximum. Our analysis relies heavily on upper and lower bounds of fractional integrals, where we establish a novel improvement of the classical Hardy--Littlewood inequality for fractional integrals of a special class of step functions.