Grant-in-Aid for Scientific Research (B).
General mathematics (including Probability theory/Statistical mathematics)
|Research Institution||KYOTO UNIVERSITY|
HINO Masanod Kyoto University, Graduate School of Informatics, Ledurer, 情報学研究科, 講師 (40303888)
KUMAGAI Takashi Kyoto Univ., Graduate School of Informatics, Associate Professor, 情報学研究科, 助教授 (90234509)
SHINODA Masato Nara Women's University, Fac.of Sci., Lecturer, 理学部, 講師 (50271044)
MATSUMOTO Hiroyuki Nagoya Univ., School of Info.and Sci., Associate Professor, 情報文化学部, 助教授 (00190538)
TAKAHASHI Yoichiro Kyoto University, Res.Inst.Math.Sci., Professor, 数理解析研究所, 教授 (20033889)
KIGAMI Jun Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (90202035)
杉浦 誠 名古屋大学, 多元数理科学研究科, 助手 (70252228)
|Project Fiscal Year
1998 – 2000
Completed(Fiscal Year 2000)
|Budget Amount *help
¥7,800,000 (Direct Cost : ¥7,800,000)
Fiscal Year 2000 : ¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1999 : ¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1998 : ¥2,400,000 (Direct Cost : ¥2,400,000)
|Keywords||Fractal / Diffusion Process / Stochastic Process / Multi-fractal / Hausdorff dimension / Heat kernel / Limit theorem / Besov space / フラクタル / 拡散過程 / 確率過程 / マルチフラクタル / ハウスドルフ次元 / 熱核 / 極限定理 / ベソフ空間 / 固有値分布 / ランダムウォーク|
We have established the following results throughout this project.
1. On spectra of self-adjoint operators on fractals
We study the short time asymptotics for heat kernels when the self-similarity of the measure and that of the diffusion process are different. In this case, the asymptotics highly depend on the initial points and there is a multi-fractality. We compute the Hausdorff dimension and the paper will appear. Further, we define a family of quasi-distances and show that the Hausdorff dimension can be expressed in a simple way using some special quasi-distance.
2. On stochastic processes on random fractals
(1) We construct a diffusion process and study its heat kernel properties on a homogeneous random Sierpinski carpet.
(2) We study a short time heat kernel asymptotics for a diffusion process on a random recursive Sierpinski gasket, which does not have spatial symmetries.
These results are in our paper that have already appeared.
3. Stochastic analysis on fractals
(1) We construct a diffusion process on a (Euclidean) space where countable numbers of disordered media (such as fractals) are embedded.
We apply a trace theory of Besov spaces for the proof. The paper has already appeared and we are now working on large deviations for the process.
(2) Concerning the "stochastic analysis for stochastic processes with rough paths" studied by T.Lyons, we have not obtained a new result so far. It is one of the future problem to study the detailed properties of the stochastic differential equations established by him.