# Seminar at SMU Delhi

January 7, 2016 (Thursday) , 3:30 PM at Webinar
Speaker: W. T. Frank den Hollander, Leiden University
Title: How porous is a Brownian Motion?
Abstract of Talk

The path of a Brownian motion on a $d$-dimensional torus run up to time $t$ is a random compact subset of the torus. In this talk we look at the geometric and spectral properties of the complement $C(t)$ of this set when $t$ tends to infinity. Questions we address are the following:

1. What is the linear size of the largest region in $C(t)$?

2. What does $C(t)$ look like around this region?

3. Does $C(t)$ have some sort of "component-structure"?

4. What are the largest capacity, largest volume and smallest principal Dirichlet eigenvalue of the "components" of $C(t)$?

We discuss both $d \geq 3$ and $d=2$, which turn out to be very different.

Based on joint work with Michiel van den Berg (Bristol), Erwin Bolthausen (Zurich) and Jesse Goodman (Auckland).