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The trace of spatial brownian motion is capacity-equivalent to the unit square
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  • Published: November 1996

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

  • Robin Pemantle1,
  • Yuval Peres2 &
  • Jonathan W. Shapiro2 

Probability Theory and Related Fields volume 106, pages 379–399 (1996)Cite this article

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Summary.

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 α-stable process approaches within ɛ 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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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA , , , , , , US

    Robin Pemantle

  2. Department of Statistic, University of California, Berkeley, CA 94720, USA, , , , , , US

    Yuval Peres & Jonathan W. Shapiro

Authors
  1. Robin Pemantle
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  2. Yuval Peres
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  3. Jonathan W. Shapiro
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Received: 8 February 1995 / In revised form: 27 July 1995

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Pemantle, R., Peres, Y. & Shapiro, J. The trace of spatial brownian motion is capacity-equivalent to the unit square. Probab Theory Relat Fields 106, 379–399 (1996). https://doi.org/10.1007/s004400050070

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  • Issue Date: November 1996

  • DOI: https://doi.org/10.1007/s004400050070

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Keywords

  • Brownian Motion
  • Lebesgue Measure
  • Process Approach
  • Constant Multiple
  • Capacity Estimate
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