# Uniform dimension results for fractional Brownian motion

### Richárd Balka

University of Washington, Seattle, USA and Alfréd Rényi Institute, Budapest, Hungary### Yuval Peres

Microsoft Research, Redmond, USA

## 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) $\mathbb P(\dim_H B(A)=(1/\alpha) \dim_H A \textrm{ for all } A\subset D)=1$,

(B) $\mathbb P(\dim_P B(A)=(1/\alpha) \dim_P A \textrm{ for all } A\subset D)=1$,

(C) $\mathbb P(\dim_P B(A)\geq (1/\alpha) \dim_H A \textrm{ for all } A\subset D)=1$.

We introduce a new concept of dimension, the modified Assouad dimension, denoted by $\dim_{MA}$. We prove that $\dim_{MA} D\leq \alpha$ 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 $\dim_{MA} D\leq \alpha$. Furthermore, if $D$ is a set defined by digit restrictions then (A) holds if and only if $\dim_{MA} D \leq \alpha$ or $\dim_H D=0$. The characterization of (A) remains open in general. We prove that $\dim_{MA} D\leq \alpha$ implies (B) and they are equivalent provided that $D$ is analytic. Let $D$ be compact, we show that (C) is equivalent to $\dim_H D\leq \alpha$. This implies that if $\dim_H D\leq \alpha$ and $\Gamma_D=\{E\subset B(D)\colon \dim_H E=\dim_P E\}$, then

In particular, all level sets of $B|_{D}$ have Hausdorff dimension zero almost surely.

## Cite this article

Richárd Balka, Yuval Peres, Uniform dimension results for fractional Brownian motion. J. Fractal Geom. 4 (2017), no. 2, pp. 147–183

DOI 10.4171/JFG/48