Uniform dimension results for fractional Brownian motion

Abstract

Kaufman's dimension doubling theorem states that for a planar Brownian motion $\{\mathbf{B}(t):t\in [0,1]\}$ we have

where $\dim$ may denote both Hausdorff dimension ${\dim }_H$ and packing dimension ${\dim }_P$. The main goal of the paper is to prove similar uniform dimension results in the one-dimensional case. Let $0<\alpha <1$ and let $\{B(t):t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $\alpha$. For a deterministic set $D\subset [0,1]$ consider the following statements:

(A) $\mathbb{P}({\dim }_{H}B(A)=(1/\alpha ){\dim }_{H}A\text{ for all }A\subset D)=1$,

(B) $\mathbb{P}({\dim }_{P}B(A)=(1/\alpha ){\dim }_{P}A\text{ for all }A\subset D)=1$,

(C) $\mathbb{P}({\dim }_{P}B(A)\ge (1/\alpha ){\dim }_{H}A\text{ for all }A\subset D)=1$.

We introduce a new concept of dimension, the modified Assouad dimension, denoted by ${\dim }_{MA}$. We prove that ${\dim }_{MA}D\le \alpha$ implies (A), which enables us to reprove a restriction theorem of Angel, Balka, Máthé, and Peres. We show that if $D$ is self-similar then (A) is equivalent to ${\dim }_{MA}D\le \alpha$. Furthermore, if $D$ is a set defined by digit restrictions then (A) holds if and only if ${\dim }_{MA}D\le \alpha$ or ${\dim }_{H}D=0$. The characterization of (A) remains open in general. We prove that ${\dim }_{MA}D\le \alpha$ implies (B) and they are equivalent provided that $D$ is analytic. Let $D$ be compact, we show that (C) is equivalent to ${\dim }_{H}D\le \alpha$. This implies that if ${\dim }_{H}D\le \alpha$ and ${\Gamma }_D=\{E\subset B(D):{\dim }_{H}E={\dim }_{P}E\}$, then $\mathbb{P}({\dim }_H(B^{-1}(E)\cap D)=\alpha {\dim }_{H}E\text{ for all }E\in {\Gamma }_D)=1.$ In particular, all level sets of $B{|}_D$ have Hausdorff dimension zero almost surely.