On the confinement property of two‐dimensional Brownian motion among poissonian obstacles
On the confinement property of two‐dimensional Brownian motion among poissonian obstacles
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泊松障碍物中二维布朗运动的约束性质
DOI：
10.1002/cpa.3160440822
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发表时间：
1991
影响因子：
3
通讯作者：
A. Sznitman
中科院分区：
文献类型：
--
作者：
A. Sznitman
考虑Rd上的随机障碍物，其是[Wd]的模型非极紧致子集K在具有恒定强度u> 0的随机泊松云的点处的平移。我们用P表示这个云的定律。假设给定一个独立的布朗运动Z.在Rd上，从原点开始，用法沃，即在进入障碍物时被杀死。如果T表示Z的进入时间。对于Wiener香肠，在时间t为Z时，W1 = Uos 2，-K。以-K为模型，已知当t趋于无穷大时：其中U在Rd的有界开子集的类21上运行，具有可忽略的边界，A（U）是U中-f A的主要Dirichlet特征值，并且I。I表示勒贝格体积。当K是任意半径的球时，Donsker-Varadhan（见[4]）使用大偏差技术证明了这一结果。一般K的情形在[8 1]中通过不同的方法得到了证明。事实上，（1）甚至被认为是举行时，障碍物缩小与t的速度不是太快，见Bolthausen [11，或[101]。常数c（d，u）由Donsker-Varadhan确定为：
Consider random obstacles on Rd, which are translates of a model nonpolar compact subset K of [Wd, at the points of a random Poisson cloud with constant intensity u> 0. We denote by P the law of this cloud. We suppose we are given an independent Brownian motion Z. on Rd, starting from the origin, with law Wo, which is killed upon entering the obstacles. If T stands for the entrance time of Z. in the obstacles and W,= Uos 2,-K, for the Wiener sausage in time t of Z. modeled on-K, it is known that as t goes to infinity: whereHere U runs over the class 21 of bounded open subsets of Rd with negligible boundary, A (U) is the principal Dirichlet eigenvalue of-f A in U, and I. I denotes Lebesgue volume. When K is a ball of arbitrary radius this result was proved by Donsker-Varadhan (see [4]), using a large deviation technique. The case of a general K was proved via a different method in [8 1. In fact (1) is even known to hold when obstacles are shrinking with t at a rate which is not too rapid; see Bolthausen [11, or [101. The constant c (d, u) was identified by Donsker-Varadhan as