An Isoperimetric Inequality For The Wiener Sausage

Let (ξ(s))s≥0 be a standard Brownian motion in d≥1 dimensions and let (Ds)s≥0 be a collection of open sets in $\R^d$. For each s, let Bs 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.