Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions

TL;DR

This paper investigates the local regularity of Gaussian wavelet series, including fractional Brownian motion, revealing three types of points and extending the analysis to multifractal cases, highlighting the uniqueness of slow points.

Contribution

It characterizes the local regularity of Gaussian wavelet series and fractional Brownian motion, introducing the concept of slow, ordinary, and rapid points, and extends these results to multifractal series.

Findings

Almost sure existence of three point types in Gaussian wavelet series

Fractional Brownian motion exhibits these three point types

Extension of properties to multifractal Gaussian wavelet series

Abstract

We study the H\"olderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional Brownian Motion. We also show that this property is satisfied for a multifractal extension of Gaussian wavelet series. Finally, we remark that the existence of slow points is specific to these functions.