Project/Area Number 
15540189

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Kansai University 
Principal Investigator 
ICHIHARA Kanji Kansai University, Faculty of Engineering, Professor, 工学部, 教授 (00112293)

CoInvestigator(Kenkyūbuntansha) 
FUKUSHIMA Masatoshi Kansai University, Faculty of Engineering, Professor, 工学部, 教授 (90015503)
KUSUDA Masaharu Kansai University, Faculty of Engineering, Professor, 工学部, 教授 (80195437)
CHIYONOBU Taizo Kwansei Gakuin University, School of Science, Associate Professor, 理工学部, 助教授 (50197638)
KURISU Tadashi Kansai University, Faculty of Engineering, Professor, 工学部, 教授 (00029159)
HIRASHIMA Yasumasa Kansai University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (80047399)

Project Period (FY) 
2003 – 2005

Project Status 
Completed (Fiscal Year 2005)

Budget Amount *help 
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)

Keywords  radial random walk / homogenous tree / large deviation / rate function / Brownian motion / spectrum / curvature / principal eigenfunction / ロバチェフスキー平面 / 基本領域 / 離散群 / 保形関数 / 測地流 / stable foliation / harmonic measure / 双曲的リーマン多様体 / 結晶格子 / 可逆的マルコフ連鎖 / spherical mean / 保型関数 / 樹木 / ランダムウォーク / 主固有関数 / 周期的マルコフ連鎖 
Research Abstract 
An infinite homogenous tree is a typical example of noneuclidean discrete spaces. We have established DonskerVaradhan's type large deviation for the pinned motions of a radial random walk on the above tree. It has been shown that the corresponding rate function is related to a new Markov chain defined through harmonic transform based on a positive principal eigenfunction for the generator associated with the original random walk. Note that the principal eigenfunction depends only on the structure of the tree. Secondly, there have discussed the same problems for Brownian motions on a class of hyperbolic Riemannian manifolds whose sectional curvature diverges to ∞ at infinity. We have succeeded in showing the uniform large deviation principle for this case. Namely the upper bound is proven to be valid for any closed subset. For the manifold the bottom of the spectrum of the negative Laplacian is discrete and the associated principal eigenfunction decays faster than in an exponential order. Thirdly, the explosion problem for a continuous time, reversible Markov chains on a countably infinite set has been discussed from the viewpoint of Dirichlet space.
