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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^d}$$\end{document}. For each s, let Bs be a ball centered at 0 with vol(Bs) = vol(Ds). We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + D_s))] \geq \mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + B_s))]}$$\end{document}, 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.