The Annals of Probability

Thick points for spatial Brownian motion: multifractal analysis of occupation measure

Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni

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Abstract

Let $\mathscr{T}(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for Brownian motion in $\mathbb{R}^3$. We prove that $\sup_{|x|\leq1}\mathscr{T}(x,r)/(r^{2}|\log r|)\rightarrow16/\pi^2$ a.s. as $r\rightarrow0$, thus solving a problem posed by Taylor in 1974. Furthermore, for any $a \in(0,16/\pi^2)$, the Hausdorff dimension of the set of “thick points” $x$ for which $\lim\sup_{r \to 0}\mathscr{T}(x,r)/(r^2|\log r|) = a$ is almost surely $2-a\pi^2 /8$; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for Brownian occupation measure. Analogous results hold for Brownian motion in any dimension $d \ge 3$. These results are related to the LIL of Ciesielski and Taylor for the Brownian occupation measure of small balls in the same way that Lévy’s uniform modulus of continuity, and the formula of Orey and Taylor for the dimension of “fast points ”are related to the usual LIL. We also show that the lim inf scaling of $\mathscr{T}(x,r)$ is quite different: we exhibit nonrandom $c_1,c_2 \ge 0$, such that $c_1 < \sup_x\lim \inf _{r \to 0}\mathscr{T}(x,y)/r^2 < c_2$ a.s. In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of “limsup type.”

Article information

Source
Ann. Probab., Volume 28, Number 1 (2000), 1-35.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160110

Digital Object Identifier
doi:10.1214/aop/1019160110

Mathematical Reviews number (MathSciNet)
MR1755996

Zentralblatt MATH identifier
1130.60311

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 28A80: Fractals [See also 37Fxx]
Secondary: 60J55: Local time and additive functionals 60F99: None of the above, but in this section 28A78: Hausdorff and packing measures

Keywords
thick points multifractal analysis occupation measure

Citation

Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer. Thick points for spatial Brownian motion: multifractal analysis of occupation measure. Ann. Probab. 28 (2000), no. 1, 1--35. doi:10.1214/aop/1019160110. https://projecteuclid.org/euclid.aop/1019160110


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