Lacal Behavior of TwoDimensional Brownian Motion and Hausdorff Measure
Project/Area Number  01540202 
Research Category 
GrantinAid for Scientific Research (C).

Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Toho University 
Principal Investigator 
SHIMURA Michio Toho University Faculty of Science Associate Professor, 理学部, 助教授 (90015868)

Project Fiscal Year 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1990 : ¥300,000 (Direct Cost : ¥300,000)
Fiscal Year 1989 : ¥300,000 (Direct Cost : ¥300,000)

Keywords  TwoDimensional Brownian Motion / TwoSided Flat Point / Critical Behavior / Hausdorff Measure / TwoDimensional Random Walk / Conditioned Limit Theorem / TwoDimensional Stable Process / 2次元Brown運動 / 両側平坦点 / 限界挙動 / Hausdorff測度 / 2次元Random Walk / 条件付き極限定理 / 2次元安定過程 / つづらおり点(meandering point) / Hqusdzff測度 / 錐形領域中の回遊(excursion) / 2次元ランダムウォク / 条件付極限定理 
Research Abstract 
1. Theme (I) Proof of existence of nontrivial twosided flat points for twodimensional Brownian motion (1) In 1989 we negatively conjectured on Taylor's problem as follows : " It would be impossible to divide a twodimensional 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 twodimensional 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 twodimensional Brownian paths there exist nontrivial twosided 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 twosided 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 twodimensional random walk conditioned to stay in cone".

Report
(4results)
Research Output
(4results)