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 {B(t):t[0,1]}\{\mathbf{B}(t): t \in [0,1]\} we have

P(dimB(A)=2dimA for all A[0,1])=1,\mathbb P(\dim \mathbf{B}(A)=2\dim A \textrm{ for all } A\subset [0,1])=1,

where dim\dim may denote both Hausdorff dimension dimH\dim_H and packing dimension dimP\dim_P. The main goal of the paper is to prove similar uniform dimension results in the one-dimensional case. Let 0<α<10 < \alpha < 1 and let {B(t):t[0,1]}\{B(t): t \in [0,1]\} be a fractional Brownian motion of Hurst index α\alpha. For a deterministic set D[0,1]D\subset [0,1] consider the following statements:

(A) P(dimHB(A)=(1/α)dimHA for all AD)=1\mathbb P(\dim_H B(A)=(1/\alpha) \dim_H A \textrm{ for all } A\subset D)=1,

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

(C) P(dimPB(A)(1/α)dimHA for all AD)=1\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 dimMA\dim_{MA}. We prove that dimMADα\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 DD is self-similar then (A) is equivalent to dimMADα\dim_{MA} D\leq \alpha. Furthermore, if DD is a set defined by digit restrictions then (A) holds if and only if dimMADα\dim_{MA} D \leq \alpha or dimHD=0\dim_H D=0. The characterization of (A) remains open in general. We prove that dimMADα\dim_{MA} D\leq \alpha implies (B) and they are equivalent provided that DD is analytic. Let DD be compact, we show that (C) is equivalent to dimHDα\dim_H D\leq \alpha. This implies that if dimHDα\dim_H D\leq \alpha and ΓD={EB(D) ⁣:dimHE=dimPE}\Gamma_D=\{E\subset B(D)\colon \dim_H E=\dim_P E\}, then

P(dimH(B1(E)D)=αdimHE for all EΓD)=1.\mathbb P(\dim_H (B^{-1}(E)\cap D)=\alpha \dim_H E \textrm{ for all } E\in \Gamma_D)=1.

In particular, all level sets of BDB|_{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