An isoperimetric inequality for the Wiener sausage

TL;DR

This paper proves a new isoperimetric inequality for the expected volume of the Wiener sausage, showing it is minimized when the shape is a ball, and demonstrates that adding drift increases this expected volume.

Contribution

The paper establishes a novel isoperimetric inequality for Wiener sausages, extending classical results to include drift effects in Brownian motion.

Findings

Expected Wiener sausage volume is minimized by spherical shapes.

Adding drift to Brownian motion increases the expected volume.

The inequality holds for all time t and dimensions d ≥ 1.

Abstract

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.