An Isoperimetric Inequality For The Wiener Sausage

Let \((\xi (s))_{s\geq 0}\) be a standard Brownian motion in \(d\geq 1\) dimensions and let \((D_s)_{s\geq 0}\) be a collection of open sets in \(\R^d\). For each \(s\), let \(B_s\) be a ball centered at 0 with \(\vol(B_s) = \vol(D_s)\). We show that \(\E[\vol(\cup_{s \leq t}(\xi(s) + D_s))] \geq \E[\vol(\cup_{s \leq t}(\xi(s) + B_s))]\), 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.