Some sample path properties of multifractional Brownian motion

TL;DR

This paper investigates the geometric properties of multifractional Brownian motion with irregular Hurst functions, providing a comprehensive analysis of its regularity and fractal dimensions, extending previous smooth-case results.

Contribution

It extends existing results on mBm to include irregular Hurst functions, characterizing sample path regularity and fractal dimensions in this more general setting.

Findings

Complete characterization of pointwise Hölder regularity.

Determination of Box and Hausdorff dimensions of the graph.

Illustrative examples demonstrating the geometry of mBm with irregular Hurst functions.

Abstract

The geometry of the multifractional Brownian motion (mBm) is known to present a complex and surprising form when the Hurst function is greatly irregular. Nevertheless, most of the literature devoted to the subject considers sufficiently smooth cases which lead to sample paths locally similar to a fractional Brownian motion (fBm). The main goal of this paper is therefore to extend these results to a more general frame and consider any type of continuous Hurst function. More specifically, we mainly focus on obtaining a complete characterization of the pointwise H\"older regularity of the sample paths, and the Box and Hausdorff dimensions of the graph. These results, which are somehow unusual for a Gaussian process, are illustrated by several examples, presenting in this way different aspects of the geometry of the mBm with irregular Hurst functions