Thick Points for Spatial Brownian Motion : Multifractal Analysis of Occupation MeasureBy

Let T (x; r) denote the total occupation measure of the ball of radius r centered at x for Brownian motion in IR 3. We prove that sup jxjj1 T (x; r)=(r 2 jlog rj) ! 16== 2 a.s. as r ! 0, thus solving a problem posed by Taylor in 1974. Furthermore, for any a 2 (0; 16== 2), the Hausdorr dimension of the set of \thick points" x for which limsup r!0 T (x; r)=(r 2 jlog rj) = a, is almost surely 2 ? aa 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 evy'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 T (x; r) is quite diierent: we exhibit non-random c 1 ; c 2 > 0, such that c 1 < sup x liminf r!0 T (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 Hausdorr dimension of random fractals ofìimsup type'.