A uniform result for the dimension of fractional Brownian motion level sets

Abstract

Let $B=\{B_t:t\ge 0\}$ be a real-valued fractional Brownian motion of index $H\in (0,1)$. We prove that the macroscopic Hausdorff dimension of the level sets ${\mathcal{L}}_x=\left\{t\in {\mathbb{R}}_{+}:B_t=x\right\}$ is, with probability one, equal to $1-H$ for all $x\in \mathbb{R}$.