1997 Fiscal Year Final Research Report Summary
Research of stochastic processes on fractals.
| Project/Area Number |
08454040
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nagoya University |
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Principal Investigator |
KUMAGAI Takashi Nagoya Univ.Graduate School of Math., Associate Professor, 大学院多元数理科学研究科, 助教授 (90234509)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIURA Makoto Nagoya University, Graduate School of Math., Research Assistant, 大学院多元数理科学研究科, 助手 (70252228)
CHIYONOBU Taizo Nagoya University, Graduate School of Math., Research Assistant, 大学院多元数理科学研究科, 助手 (50197638)
OBATA Nobuaki Nagoya University, Graduate School of Math., Associate Professor, 大学院多元数理科学研究科, 助教授 (10169360)
ICHIHARA Kanji Nagoya University, Graduate School of Math., Associate Professor, 大学院多元数理科学研究科, 助教授 (00112293)
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| Project Period (FY) |
1996 – 1997
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| Keywords | fractal / stochastic process / self-similar set / heat kernel / sierpinski carpet / Harnack ineguality / homogenization |
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| Research Abstract |
1. We have obtained sharp estimates on the transition densities (heat kernels) for diffusion processes on p.c.f. self-similar sets, which correspond to finitely ramified self-similar fractals. It was known that if the fractal had a strong symmetry, then the heat kernel of the Brownian motion had Aronson type estimates. In our result, we show that the Aronson type estimates do not hold in general. This work will appear in J.London Math.Soc.
2. On infinitely ramified fractals, we have studied the heat kernel estimates for diffusion processes on random Sierpinski carpets. We obtained sharp esimates for each sample carpets (each environments). Further, we obtained almost sure estimates assuming strong ergodicity for the randomness of the carpets. One of the key idea was to obtain uniform Harnack inequality of the approximate processes using the coupling arguments due to Barlow-Bass. This work is now a preprint.
3. On the relations between fractals and Euclidean spaces, we studied homogenization problems. Since the joint work of the head investigator with Prof.Kusuoka, it was known that the stability of fixed points of the renormalization map was essential. In our research, we discussed with researchers of the same fields when we attended interational workshops and learned several new ideas and methods to search for the problem. But so far we could not apply the methods to our cases. This is the problem we should pursue in a near future.
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