Asymptotics for the Wiener sausage among Poissonian obstacles

TL;DR

This paper studies the behavior of the Wiener sausage in a Poissonian obstacle environment, deriving probabilistic limits and asymptotics that reveal how Brownian motion is confined and interacts with obstacles.

Contribution

It provides new asymptotic results and large deviation principles for the Wiener sausage among Poissonian obstacles, including volume moment asymptotics and confinement behavior.

Findings

Brownian motion is confined in a ball near its starting point when surviving among obstacles

Established a weak law of large numbers and large deviation principles for the Wiener sausage

Derived moment asymptotics for the volume of the Wiener sausage

Abstract

We consider the Wiener sausage among Poissonian obstacles. The obstacle is called hard if Brownian motion entering the obstacle is immediately killed, and is called soft if it is killed at certain rate. It is known that Brownian motion conditioned to survive among obstacles is confined in a ball near its starting point. We show the weak law of large numbers, large deviation principle in special cases and the moment asymptotics for the volume of the corresponding Wiener sausage. One of the consequence of our results is that the trajectory of Brownian motion almost fills the confinement ball.