An isoperimetric inequality for the Wiener sausage

Abstract

Let (ξ(s)) s ≥ 0 be a standard Brownian motion in d ≥ 1 dimensions and let (D s ) s ≥ 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 \({\mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + D_s))] \geq \mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + B_s))]}\), 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.