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

Let F(x, r) denote the total occupation measure of the ball of radius r cen tered at x for Brownian motion in R-3. We prove that sup(\x\less than or eq ual to 1) F(x, r)/(r(2) \log r\) --> 16/pi(2) a.s. as r --> 0, thus solving a problem posed by Taylor in 1974. Furthermore, for any a epsilon (0, 16/p i(2)), the Hausdorff dimension of the set of "thick points" x for which lim sup(r-->0) F(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" fo r Brownian occupation measure. Analogous results hold for Brownian motion i n any dimension d > 3. These results are related to the LIL of Ciesielski a nd Taylor for the Brownian occupation measure of small balls in the same wa y that Levy's uniform modulus of continuity, and the formula of Grey and Ta ylor for the dimension of "fast points" are related to the usual LIL. We al so show that the lim inf scaling of F(x, r) is quite different: we exhibit nonrandom c(1), c(2) > 0, such that c(1) < sup(x) lim inf(r-->0) F(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 "l imsup type".