THE TRACE OF SPATIAL BROWNIAN-MOTION IS CAPACITY-EQUIVALENT TO THE UNIT SQUARE

We show that with probability 1, the trace B[0, 1] of Brownian motion in space, has positive capacity with respect to exactly the same kerne ls as the unit square. More precisely, the energy of occupation measur e on B[0, 1] in the kernel f(\x - y\), is bounded above and below by c onstant multiples of the energy of Lebesgue measure on the unit square . (The constants are random, but do not depend on the kernel.) As an a pplication, we give almost-sure asymptotics for the probability that a n alpha-stable process approaches within epsilon of B[0, 1], condition al on B[0, 1]. The upper bound on energy is based on a strong law for the approximate self-intersections of the Brownian path. We also prove analogous capacity estimates for planar Brownian motion and for the z ero-set of one-dimensional Brownian motion.