Some sample path properties of multifractional Brownian motion

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