An isoperimetric inequality for the Wiener sausage

Abstract

Let $(\xi (s){)}_{s\ge 0}$ be a standard Brownian motion in $d\ge 1$ dimensions and let $(D_s{)}_{s\ge 0}$ be a collection of open sets in ${\mathbb{R}}^d$. For each $s$, let $B_s$ be a ball centered at 0 with $\vol(B_s) = \vol(D_s)$. We show that $\E[\vol(\cup_{s \leq t}(ξ(s) + D_s))] \geq \E[\vol(\cup_{s \leq t}(ξ(s) + B_s))]$, for all $t$. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.