| Project/Area Number |
01540202
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Toho University |
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Principal Investigator |
SHIMURA Michio Toho University Faculty of Science Associate Professor, 理学部, 助教授 (90015868)
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| Project Period (FY) |
1989 – 1990
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| Project Status |
Completed (Fiscal Year 1990)
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| 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)
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| Keywords | 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次元ランダムウォ-ク / 条件付極限定理 |
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| Research Abstract |
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".
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