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The trace of spatial Brownian motion is capacity-equivalent to the unit square

Robin Pemantle·Yuval Peres·Jonathan W. Shapiro
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Abstract

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.

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Robin Pemantle, Yuval Peres, Jonathan W. Shapiro. 2004-04-05. The trace of spatial Brownian motion is capacity-equivalent to the unit square. https://arxiv.org/abs/math/0404088

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