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 Rd.
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