Abstract: The path $W[0,t]$ of a Brownian motion on a $d$-dimensional torus $T^d$ run for time $t$ is a random compact subset of $T^d$. In this talk we look at the geometric properties of the complement $C(t) = T^d\W[0,t]$ as $t\to\infty$ for $d\geq 3$. 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 speculate about what happens for $d=2$, which is much harder to understand. Joint work with Jesse Goodman (Haifa)