2000 Fiscal Year Final Research Report Summary
Research on vibrations and diffusions on fractals
| Project/Area Number |
11837008
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
KIGAMI Jun Professor, Graduate School of Informatics, Kyoto University, 情報学研究科, 教授 (90202035)
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| Co-Investigator(Kenkyū-buntansha) |
KUMAGAI Takashi Associate professor, Graduate School of Informatics, Kyoto University, 情報学研究科, 助教授 (90234509)
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| Project Period (FY) |
1999 – 2000
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| Keywords | fractals / self-similar sets / Laplacian / diffusion / Dirichlet form / heat kernel / asymptotic behavior |
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| Research Abstract |
In this research project, we obtained the following five main results related with analysis on fractals.
(1) Markov property of Dirichlet forms on self-similar sets
We showed the Markov property of Kusuoka-Zhou's Dirichlet forms on self-similar sets.
(2) Self-similarity of volume measures associated with Laplacians on p.c.f. self-similar fractals
We obtained a sufficient condition for the self-similarity of the volume measure, which is defined by using operator theoretic trace. We also showed that the sufficient condition holds in the case of standard Laplacian on the Sierpinski gasket.
(3) Green's function on fractals
We obtained an algorithm to calculate the diagonal of Green's function and used the algorithm to investigate the maximum value of Green's function.
(4) Large deviations for Brownian motion on the Sierpinski gasket
We showed that Varadhan type estimate and Schilder type Large deviation do not hold for the case of the standard Laplacian on the Sierpinski gasket
(5) Multifractal formalisms for the local spectral and walk dimensions
We showed the multifractal nature of the local spectral and walk dimensions associated with the Laplacians on the self-similar sets.
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