Torsional Rigidity for Regions with a Brownian Boundary

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 W₁[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.