Moderate deviations for the volume of the Wiener sausage

By M. Berg, E. Bolthausen, Frank Den Hollander. 2001

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For a > 0, let Wa(t) be the a-neighbourhood of standard Brownian motion in Rd starting at 0 and observed until time t. It is well-known that ElWa(t)l IKat (t -> oc) for d > 3, with Kea the Newtonian capacity of the ball with radius a. We prove that lm t(d- 2)/d log p(Wa(t)L oc. This is markedly different from the optimal strategy for large deviations {IWa(t)I 5 is nonanalytic at some critical point in (0, Kia), above which it follows a pure power law. This crossover is associated with a collapse transition in the optimal strategy. We also derive the analogous moderate deviation result for d = 2. In this case EIWa(t) 2wrt/logt (t -> oc), and we prove that

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eae1ea972b64b7c2bada71fe4f9065e84956f4c7

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