Project/Area Number  08454040 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Nagoya University 
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)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1996 – 1997

Project Status 
Completed(Fiscal Year 1997)

Budget Amount *help 
¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1997 : ¥1,800,000 (Direct Cost : ¥1,800,000)

Keywords  fractal / stochastic process / selfsimilar set / heat kernel / sierpinski carpet / Harnack ineguality / homogenization / フラクタル / 確率過程 / 自己相似集合 / 熱核 / シェルピンスキーカーペット / ハルナック不等式 / ホモジナイゼーション / 拡散過程 / ランダムウォーク / ハウスドルク次元 / 固有値分布 
Research Abstract 
1. We have obtained sharp estimates on the transition densities (heat kernels) for diffusion processes on p.c.f. selfsimilar sets, which correspond to finitely ramified selfsimilar 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 BarlowBass. 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.
