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.<\/p>\n","protected":false},"excerpt":{"rendered":"
Let be a standard Brownian motion in dimensions and let be a collection of open sets in . For each , let be a ball centered at 0 with . We show that , for all . 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