Lacal Behavior of Two-Dimensional Brownian Motion and Hausdorff Measure

二维布朗运动的局部行为和豪斯多夫测度

• 批准号: 01540202
• 负责人: SHIMURA Michio
• 金额: $ 0.38万
• 依托单位: Toho University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1989
• 资助国家: 日本
• 起止时间: 1989 至 1990
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-01540202/
• 关键词: Two-Dimensional Brownian Motion; Two-Sided Flat Point; Critical Behavior; Hausdorff Measure; Two-Dimensional Random Walk; Conditioned Limit Theorem; Two-Dimensional Stable Process; つづらおり点(meandering point); Hqusdzff測度; 錐形領域中の回遊(excursion); 2次元ラン

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".

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 certainly. (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泰勒的问题和二维布朗运动的关键行为之一（ii）找到一组双面平坦点的确切的Hausdorff测量函数，它从主题证明（i）中遵循以下概念：考虑hausdorff的测量函数，以便1/[logx]^r（r> 0）为x->+0。然后，我们将拥有以下内容的R_0> 0：集合的Hausdorff测量为 *或0为0 <r <r_0或r_0。我们将证明这个概念。 3。其他结果我们完成了一篇题为“二维随机步行的限制定理，以保持圆锥状态”。

项目成果

Michio Shimura: "A limit theorem for two-dimensional random walk conditioned to stay in a cone" Yokohama Mathematical J.Vol. 39. (1991)

志村道雄：“以圆锥体为条件的二维随机游走的极限定理”横滨数学杂志卷。

Michio Shimura: "A limit theorem for twoーdimensional random walk couditioned to stay in a cone" Yokohama Mathematical Journal. 39. (1991)

Michio Shimura：“二维随机游走的极限定理可保持在圆锥体中”，横滨数学杂志 39。（1991）

Michio Shimura: "A limit theorew for twoーdimensional random walk conditioned to stay in a cone" Yokohama Mathematical Journal. 39. (1991)

Michio Shimura：“二维随机游走的极限理论，条件为保持在圆锥体中”，横滨数学杂志 39。（1991）