Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni
\def \rar{\rightarrow} \def \TT{{\mathcal T}} \def \reals{{\mathbb R}} Let $\TT(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for Brownian motion in $\reals^3$. %, and write $z(x,r)= \TT(x,r)/(r^2|\log r|)$. We prove that $\sup_{|x| \leq 1} \TT(x,r)/(r^2|\log r|) \to 16/\pi^2$ a.s. %JJ as $r \rar 0$, 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 $\limsup_{r \rightarrow 0} \TT(x,r)/(r^2|\log r|)=a$, 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 $d > 3$. These results are related to the LIL of Ciesielski and Taylor (1962) for the Brownian occupation measure of small balls, in the same way that L\'{e}vy's uniform modulus of continuity, and the formula of Orey and Taylor (1974) for the dimension of ``fast points'', are related to the usual LIL. We also show that the $\liminf$ scaling of $\TT(x,r)$ is quite different: we exhibit non-random $c_1,c_2>0$, such that $c_1 < \sup_x \liminf_{r \to 0} \TT(x,r)/r^2 < c_2 \; \,$ 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'.