| Project/Area Number |
12640125
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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 Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nara Women's University |
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Principal Investigator |
TOMISAKA Matsuyo Nara Women's University, Faculty of Sciences, Professor, 理学部, 教授 (50093977)
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| Co-Investigator(Kenkyū-buntansha) |
SHINODA Masato Nara Women's University, Faculty of Sciences, Lecturer, 理学部, 講師 (50271044)
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| Project Period (FY) |
2000 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | non-local Dirichlet form / diffusion process / time changed process / skew product / Feller property / ディリクレ形式 / bi-generalized diffusion / distorted Brownian motion / スペクトル / パーコレーション / シェルピンスキーカーペット / 相転移現象 / distorted Brownian Motion / 有限次元分布の収束 / フェラー過程 |
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| Research Abstract |
We are concerned with a sequence of diffusion processes whose underlying measures of Dirichlet forms converge to a degenerate one. Especially, we are interested in the case where a sequence of diffusion processes converges to a non-local Markov process, which never happens unless the underlying measures degenerate. As a prototype of such sequences, we adopt that of diffusion processes, which are skew products of finite dimensional diffusion processes and one-dimensional ones. This enables us to express the semigroups as time changed processes of product diffusions and obtain some properties such as convergence of semigroups and Feller property of the limit process. The feature of the Dirichlet forms with degenerate underlying measures and Dirichlet energy are already known and it is described by means of harmonic operators. We would notice that those are actually expressed as a sum of local Dirichlet forms on the support of the underlying measure and non-local Dirichlet forms on its boundaries. We will give the explicit form of the Levy measure for the Dirichlet form obtained as a limit of diffusion processes in our setting.
The Dirichlet form of the type thus obtained corresponds to the second order differential equation with non-local type boundary condition. Our simpler proof of Feller property here offers another approach to this problem. However we must confess that this is fully based on the skew product structure, and the rather tedious real analysis might be on stage if one deal with more general processes. Finally, we would say that the results even in our case give a new type of phenomena of the convergence of Dirichlet forms and well provide the rough shape of formulas in forthcoming general settings.
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