Konferenzvortragsdetails

Konferenz: Workshop on the occasion of Erwin Bolthausen's 70th birthday


Torsional rigidity of Brownian motion on the torus

Vortrag von Prof. Dr. Frank den Hollander
Datum: 16.09.16   Zeit: 15.30 - 16.20   Raum: Y16G05

We consider a Brownian motion $\beta[0,t]$ on the $m$-dimensional unit torus up to time $t$. We compute the leading order asymptotics of the expected time an independent Brownian motion $\beta'$ takes until it hits $\beta[0,t]$, in the limit as $t \to \infty$ when both $\beta$ and $\beta'$ start randomly. For $m=2$ the main contribution comes from the components whose inradius is comparable to the largest inradius, while for $m=3$ most of the torus contributes. A similar result holds for $m \geq 4$ after the Brownian motion $\beta[0,t]$ is replaced by a Wiener sausage $W_{r(t)}[0,t]$ of radius $r(t)=o(t^{-1/(m-2)})$, provided $r(t)$ decays slowly enough to ensure that the expected time tends zero. Asymptotic properties of the capacity of $\beta[0,t]$ and $W_1[0,t]$ play a central role. Our results contribute to a better understanding of the geometry of the complement of $\beta[0,t]$, which has received a lot of attention in the literature in past years.

Joint work with E. Bolthausen and M. van den Berg.