Abstract
Let T^{m} be the mdimensional 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 T^{2}\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of T^{3}\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/(m2)}), 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 ℝ^{3} 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.
Original language  English 

Pages (fromto)  375403 
Number of pages  29 
Journal  Potential Analysis 
Volume  48 
Issue number  3 
Early online date  7 Jul 2017 
DOIs  
Publication status  Published  1 Apr 2018 
Keywords
 Brownian motion
 Capacity
 Heat kernel
 Inradius
 Laplacian
 Spectrum
 Torsional rigidity
 Torus
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Professor Michiel van den Berg
 School of Mathematics  Emeritus Professor
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member, Honorary and Visiting Academic