Replica overlap and covering time for the Wiener sausages among Poissonian obstacles

TL;DR

This paper investigates the behavior of Wiener sausages among Poissonian obstacles, showing that their intersection asymptotically equals their union and that the covering time scales with the volume of the confined region.

Contribution

It provides new asymptotic results on replica overlap and covering time for Wiener sausages in media with Poissonian obstacles, confirming strong localization effects.

Findings

Replica overlap asymptotically equals the union of two Wiener sausages.

Covering time scales with the volume of the confinement region.

Particles are strongly localized by the media.

Abstract

We study two objects concerning the Wiener sausage among Poissonian obstacles. The first is the asymptotics for the \textit{replica overlap}, which is the intersection of two independent Wiener sausages. We show that it is asymptotically equal to their union. This result confirms that the localizing effect of the media is so strong as to completely determine the motional range of particles. The second is an estimate on the \textit{covering time}. It is known that the Wiener sausage avoiding Poissonian obstacles up to time $t$ is confined in some `clearing' ball near the origin and almost fills it. We prove here that the time needed to fill the confinement ball has the same order as its volume.