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


Kaufman's dimension doubling theorem states that for a planar Brownian motion we have

where may denote both Hausdorff dimension and packing dimension . The main goal of the paper is to prove similar uniform dimension results in the one-dimensional case. Let and let be a fractional Brownian motion of Hurst index . For a deterministic set consider the following statements:

(A) ,

(B) ,

(C) .

We introduce a new concept of dimension, the modified Assouad dimension, denoted by . We prove that implies (A), which enables us to reprove a restriction theorem of Angel, Balka, Máthé, and Peres. We show that if is self-similar then (A) is equivalent to . Furthermore, if is a set defined by digit restrictions then (A) holds if and only if or . The characterization of (A) remains open in general. We prove that implies (B) and they are equivalent provided that is analytic. Let be compact, we show that (C) is equivalent to . This implies that if and , then In particular, all level sets of 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