On the volume of the intersection of two Wiener sausages
On the volume of the intersection of two Wiener sausages
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DOI：
10.4007/annals.2004.159.741
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发表时间：
2004-03
影响因子：
4.9
通讯作者：
Michel van den Berg;E. Bolthausen;Frank den Hollander
中科院分区：
文献类型：
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作者：
Michel van den Berg;E. Bolthausen;Frank den Hollander
对于> 0，设wa1 (t)和wa2 (t)是R d中两个独立的标准布朗运动的a邻域，从0开始，观察到时间t。我们证明，对于d3和c> 0, lim t！1 1 t (d2)=d logP jwa2 (ct) \ wa2 (ct) jt = iaa d (c)并推导出速率常数iaa d (c)的变分表示。这里，a是半径为a的球的牛顿容量。我们表明，实现上述大偏差的最佳策略是让wa1 （ct）和wa2 （ct）“形成一个瑞士奶酪”：两个维也纳香肠覆盖了部分空间，留下随机的洞，它们的大小为1阶，密度根据某个最优曲线在尺度t1 =d上变化。我们详细研究了c7函数！我是d (c)。结果是I a d (c) =
For a > 0, let W a 1 (t) and W a 2 (t) be the a-neighbourhoods of two independent standard Brownian motions in R d starting at 0 and observed until time t. We prove that, for d 3 and c > 0, lim t!1 1 t (d 2)=d logP jW a 1 (ct) \W a 2 (ct)j t = I a d (c) and derive a variational representation for the rate constant I a d (c). Here, a is the Newtonian capacity of the ball with radius a. We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to \form a Swiss cheese": the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1=d according to a certain optimal prole. We study in detail the function c 7! I a d (c). It turns out that I a d (c) =