Torsional rigidity for regions with a Brownian boundary

TL;DR

This paper analyzes the asymptotic behavior of the torsional rigidity of regions in a torus with Brownian boundaries, revealing how the geometry of the complement evolves over time for different dimensions.

Contribution

It provides the first detailed asymptotic analysis of torsional rigidity for regions with Brownian boundaries on tori across various dimensions.

Findings

For $m=2$, main contribution from largest inradius components.

For $m=3$, most of the torus contributes to torsional rigidity.

Results depend on the capacity of Brownian paths and Wiener sausages.

Abstract

Let $T^m$ be the $m$-dimensional unit torus, $m \in N$. The torsional rigidity of an open set $\Omega \subset T^m$ is the integral with respect to Lebesgue measure over all starting points $x \in \Omega$ of the expected lifetime in $\Omega$ of a Brownian motion starting at $x$. In this paper we consider $\Omega = T^m \backslash \beta[0,t]$, the complement of the path $\beta[0,t]$ 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 $t \to \infty$. For $m=2$ the main contribution comes from the components in $T^2 \backslash \beta [0,t]$ whose inradius is comparable to the largest inradius, while for $m=3$ most of $T^3 \backslash \beta [0,t]$ contributes. A similar result holds for $m \geq 4$ 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 $\beta[0,t]$ in $R^3$ and $W_1[0,t]$ in $R^m$, $m \geq 4$, 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.