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An isoperimetric inequality for the Wiener sausage

By Yuval Peres and Perla Sousi


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

Topics: Mathematics - Probability, Mathematics - Functional Analysis, Primary 26D15, 60J65, Secondary 60D05, 60G55, 60G50
Year: 2011
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