{"id":357464,"date":"2017-01-25T10:27:41","date_gmt":"2017-01-25T18:27:41","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357464"},"modified":"2018-10-16T19:59:32","modified_gmt":"2018-10-17T02:59:32","slug":"isoperimetric-inequality-wiener-sausage","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/isoperimetric-inequality-wiener-sausage\/","title":{"rendered":"An Isoperimetric Inequality For The Wiener Sausage"},"content":{"rendered":"

Let (<\/span>\u03be<\/span>(<\/span>s<\/span>)<\/span>)<\/span>s<\/span>\u2265<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span> be a standard Brownian motion in d<\/span>\u2265<\/span>1<\/span><\/span><\/span><\/span> dimensions and let (<\/span>D<\/span>s<\/span><\/span>)<\/span>s<\/span>\u2265<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span> be a collection of open sets in $\\R^d$<\/span><\/span><\/span>. For each s<\/span><\/span><\/span><\/span>, let B<\/span>s<\/span><\/span><\/span><\/span><\/span> be a ball centered at 0 with $\\vol(B_s) = \\vol(D_s)$<\/span><\/span><\/span>. We show that $\\E[\\vol(\\cup_{s \\leq t}(\\xi(s) + D_s))] \\geq \\E[\\vol(\\cup_{s \\leq t}(\\xi(s) + B_s))]$<\/span><\/span><\/span>, for all t<\/span><\/span><\/span><\/span>. 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 (\u03be(s))s\u22650 be a standard Brownian motion in d\u22651 dimensions and let (Ds)s\u22650 be a collection of open sets in $\\R^d$. 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, […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cornell University 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