The exact Hausdorff measure of the sample path for planar Brownian motion
The exact Hausdorff measure of the sample path for planar Brownian motion
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平面布朗运动样本路径的精确豪斯多夫测度
DOI：
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发表时间：
1964
影响因子：
0.8
通讯作者：
Stephen Taylor
中科院分区：
文献类型：
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作者：
Stephen Taylor
它早已被称为（见Lévy（3），页。256，260），平面上布朗运动的样本路径形成一个处处稠密的零勒贝格测度集，概率为1。在（7）中，一个容量参数被用来证明关于tα的豪斯多夫测度是无穷大的，0 < α < 2，概率为1，因此路集的维数已知为2。如果考虑样本路径的初始部分Cω = C（1，ω），0 ≤ t ≤ 1，那么询问是否存在一个测度函数ψ（t）使得，对于适当的正常数c1，c2，概率为1，变得有趣。解决了k-空间（k ≥ 3）中路的相应问题。在这种情况下，如果φ1（t）= t2 log log t−1，Lévy（4）得到了上界，Ciesielski和Taylor（1）得到了下界。对于平面情形，路径是递归的，复杂的精细结构使得测度函数φ1（t）不合适。在（2）中，ErdIgnand Taylor证明了测度对于φ2（t）= t2 log t−1以概率1是有限的，当时我们认为（1）对于φ 2（t）= φ2（t）可能是真的。最近Ray（5）得到了（1）中的下界，本文的目的是得到（1）中相同测度函数的上界，从而说明（2）定义了测度平面布朗运动的正确测度函数。
It has long been known (see Lévy (3), pp. 256, 260) that the sample paths of Brownian motion in the plane form an everywhere dense set of zero Lebesgue measure, with probability 1. In (7), a capacity argument was used to show that the Hausdorff measure with respect to tα is infinite for 0 < α < 2 with probability 1 so that the dimension of the path set is known to be 2. If one considers the initial part of the sample path Cω = C(1, ω) for 0 ≤ t ≤ 1, then it becomes interesting to ask if there is a measure function ψ(t) such that, with probability 1, for suitable positive constants c1, c2. The corresponding problem for paths in k-space (k ≥ 3) has been solved. In this case, if φ1(t) = t2 log log t−1, Lévy (4) obtained the upper bound and Ciesielski and Taylor (1) obtained the lower bound. For the planar case, the path is recurrent, and the intricate fine structure makes the measure function φ1(t) inappropriate. In (2), Erdő and Taylor showed that the measure is finite with probability 1 with respect to φ2(t) = t2 log t−1, and at that time we thought that (1) might be true with ψ(t) = φ2(t). Recently Ray (5) has obtained the lower bound in (1)with The purpose of the present note is to obtain the upper bound in (1) for the same measure function, thus showing that (2) defines the correct measure function for measuring planar Brownian motion.