Moderate deviations for the volume of the Wiener sausage
Moderate deviations for the volume of the Wiener sausage
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DOI：
10.2307/2661345
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发表时间：
2001-03
期刊：
影响因子：
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通讯作者：
M. Berg;E. Bolthausen;F. Hollander
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文献类型：
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作者：
M. Berg;E. Bolthausen;F. Hollander
对于> 0，设Wa(t)为Rd中从0开始并观察到时间t的标准布朗运动的a邻域。众所周知，对于d > 3, ElWa(t) 1 IKat (t -> oc)，其中Kea为半径为a的球的牛顿容量。我们证明了lm t(d- 2)/d log p(Wa(t) l oc。这与大偏差的最佳策略明显不同{IWa(t) i5在（0,Kia）的某个临界点是非解析的，在此之上它遵循纯粹的幂律。这种交叉与最优策略中的崩溃过渡有关。我们也得到了d = 2时类似的中等偏差结果。这里EIWa(t) 2wrt/logt （t - >oc）我们证明了它
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