  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 kernels as the
unit square. More precisely, the energy of occupation measure on B[0,1] in the
kernel f(|x-y|), is bounded above and below by constant multiples of the energy
of Lebesgue measure on the unit square. (The constants are random, but do not
depend on the kernel.) As an application, we give almost-sure asymptotics for
the probability that an alpha-stable process approaches within epsilon of
B[0,1], conditional 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 zero-set of
one-dimensional Brownian motion.
