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

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.”