Torsional rigidity for regions with a Brownian boundary

Autor: Erwin Bolthausen, van den M Berg, den WThF Frank Hollander
Rok vydání: 2017
Předmět:
Zdroj: Potential Analysis, 48(3), 375-403
van den Berg, M, Bolthausen, E & den Hollander, F 2018, ' Torsional Rigidity for Regions with a Brownian Boundary ', Potential Analysis, vol. 48, no. 3, pp. 375-403 . https://doi.org/10.1007/s11118-017-9640-z
Potential Analysis
DOI: 10.1007/s11118-017-9640-z
Popis: 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.
Comment: 26 pages, 1 figure
Databáze: OpenAIRE
Externí odkaz: