Lacal Behavior of Two-Dimensional Brownian Motion and Hausdorff Measure

1.主题（I）两个维度布朗运动的非琐事两个平面点的存在证明（1）在1989年，我们对泰勒的问题进行了消极的怀疑：“这将是不可能的，将两个维度的布朗路径划分成两条随机直线，几乎完全正确的”。（这是1990年霍什尼维桑实际上提出的。）我们也有意见，因为我们可能对主题有一个积极的答案（I）是泰勒问题的变体，也是两个维度布朗运动的关键行为之一。然后我们有一个关于主题证明的大纲（一）。（2）在1990年，我们完成了主题证明（一）。Theorem we got there was as follows ：“For almost Sure two dimensional Brownian paths there exist non-trivial two sided flat point， from which we may find point as close to Taylor's one as we wish。“2.”主题（II）找到一套精确的Hausdorff度量函数，用于两个平面点，它从主题的证明（I）我们有下面的概念：考虑Hausdorff度量函数，其中1/[logx]^r （r>0）等于x->+0。然后我们将有r_0&gt;0用于后面的持有者：豪斯多夫的度量将是 * 或0根据0&lt;<r_0 or r>r_0。我们将展示我们的愿景。3.其他结果“我们的纸张entitled”限制theorem for two-dimensional random walk conditioned to stay in cone”。

1. Theme (I) Proof of existence of non-trivial two-sided flat points for two-dimensional Brownian motion (1) In 1989 we negatively conjectured on Taylor's problem as follows : " It would be impossible to divide a two-dimensional Brownian path into two pieces by a random straight line almost surely. (It was actually proved by Khoshnevisan in 1990.) We also had the opinion that we might have a positive answer to Theme (I) which was a variation to Taylor's problem and one of the critical behaviors of two-dimensional Brownian motion. Then we had an outline of a proof of Theme (I). (2) In 1990 we completed the proof of Theme (I). Theorem we got there was as follows : " For almost sure two-dimensional Brownian paths there exist non-trivial two-sided flat points, from which we may find points as close to Taylor's one as we wish. " 2. Theme (II) Find the exact Hausdorff measure function for a set of two-sided flat points It follows from the proof of Theme (I) we had the following conjecture : Consider a Hausdorff measure function such that 1/[logx]^r (r>0) as xー>+0. Then we would have r_0>0 for which the following holds : The Hausdorff measure of the set would be * or 0 according as 0<r<r_0 or r>r_0. We will prove the conjecture. 3. Other results We completed a paper entitled " A limit theorem for two-dimensional random walk conditioned to stay in cone".