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Potential Analysis

, Volume 48, Issue 3, pp 375–403 | Cite as

Torsional Rigidity for Regions with a Brownian Boundary

  • M. van den Berg
  • E. Bolthausen
  • F. den Hollander
Open Access
Article

Abstract

Let 𝕋 m be the m-dimensional unit torus, m ∈ ℕ. The torsional rigidity of an open set Ω ⊂ 𝕋 m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = 𝕋 m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in 𝕋2\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of 𝕋3\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W r(t)[0, t] of radius r(t) = o(t -1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ℝ3 and W 1[0, t] in ℝ m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on 𝕋 m , which has received a lot of attention in the literature in past years.

Keywords

Torus Laplacian Brownian motion Torsional rigidity Inradius Capacity Spectrum Heat kernel 

Mathematics Subject Classification (2010)

35J20 60G50 

Notes

Acknowledgments

The authors acknowledge support by The Leverhulme Trust through International Network Grant Laplacians, Random Walks, Bose Gas, Quantum Spin Systems. EB is supported by SNSF-grant 20-100536/1. FdH is supported by ERC Advanced Grant 267356-VARIS and NWO Gravitation Grant 024.002.003-NETWORKS. The authors thank Greg Lawler for helpful discussions about capacity of the Wiener sausage.

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Institut für MathematikUniversität ZürichZürichSwitzerland
  3. 3.Mathematical InstituteLeiden UniversityLeidenThe Netherlands

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