Torsional Rigidity for Regions with a Brownian Boundary
Let Tm be the m dimensional unit torus, m N. The torsional rigidity of an open set O Tm is the integral with respect to Lebesgue measure over all starting points x O of the expected lifetime in O of a Brownian motion starting at x. In this paper we consider O Tm\b[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\b0, t] whose inradius is comparable to the largest inradius, while for m 3 most of T3\b[0, t] contributes. A similar result holds for m 4 after the Brownian path is replaced by a shrinking Wiener sausage Wr(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 Rm, 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 Tm, which has received a lot of attention in the literature in past years.