Thin points for Brownian motion

Abstract

Let Θ( x, r) denote the occupation measure of the ball of radius r centered at x for Brownian motion { W t } 0≤ t≤1 in R d, d≥2 . We prove that for any analytic set E in [0,1], we have inf t∈E lim inf r→0 Θ(W t,r)/(r 2/| logr|)=1/ dim P (E) , where dim P ( E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which lim inf r→0 Θ(x,r)/(r 2/| logr|)=a , is almost surely 2−2/ a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of `limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.