一维布朗运动样本图的维数性质

Dimensional Properties of the Graph of 1-dimensional Brownian Motion

本文证明了一维布朗运动样本图的一条维数性质，即如果Ｘ：［０，∞］→Ｒ是布朗运动，则除去一零概率集以外，对任意集合，Ｘ在Ｆ＋ｔ上的图ＧｒａｐｈＸ（Ｆ＋ｔ）的Ｈａｕｓｄｏｒｆｆ维数ｄｉｍＨＧｒａｐｈＸ（Ｆ＋ｔ）＝ｍｉｎ（２ｄｉｍＨＦ，ｄｉｍＭＦ）对几乎所有的ｔ＞０成立．

In this paper,We prove a dimensional property of the sample graph of 1-dimensional Brownian motion,i. e,let X be 1-dimensional Brownian motion,then except a set with probability 0,for each ,dimHGraphX (F+t ) =1,min (2dimHF,dimHF) for a.et>0.