Project/Area Number 
11640163

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Nagoya University 
Principal Investigator 
ICHIHARA Kanji Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (00112293)

CoInvestigator(Kenkyūbuntansha) 
MIYAKE Masatake Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70019496)
CHIYONOBU Taizo Nagoya University, Graduate School of Mathematics, Research Associate, 大学院・多元数理科学研究科, 助手 (50197638)
OSADA Hirofumi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20177207)
SUGIURA Makoro Ryukyu University, Focalty of Science, Associate Professor, 理学部, 助教授 (70252228)

Project Period (FY) 
1999 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)

Keywords  heat kernel / harmonic transform / curvature / pinned process / large deviation / covering space / Gibbs measure / Serpinski carpet / 負曲率リーマン多様体 / ラプラシアン / スペクトル / ブラウン運動 / Labachevsky空間 / pinned process 
Research Abstract 
Ichihara has studied about some relationships between the spectrums of a second order elliptic partial differential operator and the asymptotic properties of the associated heat kernel. First, the exact decay order of the heat kernels in time has been given for a class of complete, simplyconnected, negatively curved Riemannian manifolds with asymptotically constant negative sectional curvature. In particular, it has been clarified that there exists an intimate connection between the decay part of polynomial order of the heat kernel and a term of the perturbation of constant negative curvature. Furthermore large deviation principles of the DonskerVaradhan type have been established for a class of diffusion processes with strong transience property. It is well known that the usual large deviation for a Markov process does not hold when the infimum of the spectrums for its infinitesimal generator ×(1) is positive. Ichihara has shown that it is still possible to get a nice large deviation principle so far as the pinned process is concerned if an appropriate rate function is chosen. Osada has constructed a Gibbs measure on the path space under the existence of interaction potential and has proved its uniquness in a class of Gibbs measures. Making use of the theory of Dirichlet spaces, a diffusion process having the above Gibbs measure as an invariant probabilty measure has been constructed on the path space. Furthermore Osada has constructed diffusion processes on the Sierpinski Carpet and has given estimates for their transition densities. Chiyonobu has carried out precise estimates for integrations of functionals in an infinite dimensional space, which appear in statistical mechanics.
