Torsional Rigidity for Regions with a Brownian Boundary

M. van den Berg, Erwin Bolthausen, Frank den Hollander · Potential Analysis · 2017

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.

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