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


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.


Torus Laplacian Brownian motion Torsional rigidity Inradius Capacity Spectrum Heat kernel 

Mathematics Subject Classification (2010)

35J20 60G50 



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.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 9th edn. Dover Publications, New York (1972)Google Scholar
  2. 2.
    Ancona, A.: On strong barriers and an inequality of Hardy for domains in ℝn. J. Lond. Math. Soc. 34, 274–290 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asselah, A., Schapira, B., Sousi, P.: Strong law of large numbers for the capacity of the Wiener sausage in dimension four. arXiv:1601.04576
  4. 4.
    Banuelos, R., van den Berg, M., Carroll, T.: Torsional rigidity and expected lifetime of Brownian motion. J. London Math. Soc. 66, 499–512 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Belius, D., Kistler, N.: The subleading order of two dimensional cover times. Probab. Theory Relat. Fields 167, 461–552 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    van den Berg, M., Bolthausen, E., den Hollander, F.: Heat content and inradius for regions with a Brownian boundary. Potential Anal. 41, 501–515 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    van den Berg, M., Gilkey, P.B.: Heat content and a Hardy inequality for complete Riemannian manifolds. Bull London Math. Soc. 36, 577–586 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    van den Berg, M., Nitsch, C., Trombetti, C., Ferone, V.: On Pólya’s inequality for torsional rigidity and first Dirichlet eigenvalue. Integr. Equ. Oper. Theory 86, 579–600 (2016)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brasco, L., De Philippis, G.: Spectral inequalities in quantitative form. arXiv:1604.05072
  10. 10.
    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems Progress in Nonlinear Differential Equations and their Applications, vol. 65. Birkhäuser Boston, Inc, Boston (2005)zbMATHGoogle Scholar
  11. 11.
    Dembo, A., Peres, Y., Rosen, J.: Brownian motion on compact manifolds: cover time and late points. Elect. J. Probab. 8, 1–14 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dembo, A., Peres, Y., Rosen, J., Zeitouni, O.: Cover times for Brownian motion and random walks in two dimensions. Ann. Math. 160, 433–464 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the polaron. Comm. Pure Appl. Math. 36, 505–528 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Goodman, J., den Hollander, F.: Extremal geometry of a Brownian porous medium. Probab. Theor. Relat. Fields 160, 127–174 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, New York (2007)zbMATHGoogle Scholar
  16. 16.
    Grigor’yan, A.: Heat Kernel and Analysis on Manifolds, AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society, Providence, RI; International Press, Boston (2009)Google Scholar
  17. 17.
    Kohler-Jobin, M. T.: Démonstration de l’inégalité isopérimétrique \(P\lambda ^{2}{j_{0}^{4}}/2\), conjecturée par Pólya et Szegö. (French) C. R. Acad. Sci. Paris Sér A-B 281, A119–A121 (1975)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kohler-Jobin, M.T.: Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique II. Cas inhomogène: une inégalité isopérimétrique entre la fréquence fondamentale d’une membrane et l’énergie d’équilibre d’un problème de Poisson. (French) Z. Angew. Math. Phys. 29, 767–776 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Maz’ya, V., Nazarov, S., Plamenevskij, B.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Basel, Birkhäuser Verlag (2000)CrossRefGoogle Scholar
  20. 20.
    McDonald, P.: Exit time moments and comparison theorems. Potential Anal. 38, 1365–1372 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ozawa, S.: The first eigenvalue of the Laplacian on two-dimensional Riemannian manifolds. Tôhoku Math. J. 34, 7–14 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics, Ann. of Math Stud, vol. 27. Princeton University Press, Princeton (1951)Google Scholar
  23. 23.
    Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic Press, New York (1978)zbMATHGoogle Scholar
  24. 24.
    Simon, B.: Functional Integration and Quantum Physics. Academic Press, New York (1979)zbMATHGoogle Scholar
  25. 25.
    Spitzer, F.: Electrostatic capacity, heat ow and Brownian motion. Wahrscheinlichkeitstheorie Verw Gebiete 3, 187–197 (1964)Google Scholar
  26. 26.
    Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  27. 27.
    Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Pisa 3, 697–718 (1976)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill Book Company, Inc., New York (1951)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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

Personalised recommendations