Project/Area Number  01302008 
Research Category 
GrantinAid for Cooperative Research (A).

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

Research Institution  Kyushu University 
Principal Investigator 
KUNITA Hiroshi Kyushu Univ. Fac. of Eng., Professor, 工学部, 教授 (30022552)

CoInvestigator(Kenkyūbuntansha) 
OGURA Yukio Saga Univ. Fuc. of Sci. and Eng., Professor, 理工学部, 教授 (00037847)
土谷 正明 金沢大学, 教養部, 教授 (50016101)
田中 洋 慶応義塾大学, 理工学部, 教授 (70011468)
KOTANI Shinichi Tokyo Univ. Fac. of Sci., Professor, 理学部, 教授 (10025463)
WATANABE Sinzo Kyoto Univ. Fac. of Sci., Professor, 理学部, 教授 (90025297)
SHIGA Tokuzo Tokyo Institute of Technology Univ. Fac. of Sci. Professor, 理学部, 教授 (60025418)
ITO Yuuji Keio Univ. Fac. of Sci. and Eng., Professor, 理工学部, 教授 (90112987)

Project Fiscal Year 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥13,500,000 (Direct Cost : ¥13,500,000)
Fiscal Year 1990 : ¥6,100,000 (Direct Cost : ¥6,100,000)
Fiscal Year 1989 : ¥7,400,000 (Direct Cost : ¥7,400,000)

Keywords  Stochastic analysis / Markov process / Limit theorems / Random environment / Gaussian process / Ergodic theory / Schrodinger operator / Stochastic control / 確率解析 / マルコフ過程 / 極限定理 / ランダム媒質 / ガウス過程 / エルゴド理論 / シュレディンガ作用素 / 確率制御 / 無限次元解析 / 抽象ウィナ空間 / 多粒子系マルコフ過程 / 測度値分枝過程 / 中心極限定理 / 加法過程 / マリアバン演算法 / 拡散過程 / ガウス型通信路 
Research Abstract 
We organized seven project teams in the first year and five ones in the second year and each team studied its own project. The belows are the main research results. 1. The research of the stochastic analysis : We analyzed the measures on the Wiener spaces. As results, we obtain the followings. (1) A probabilistic interpretation of the asymptotic behaviors of the fundamental solutions of a heat equation as t 0. (2) The structure of the Sobolev space in infinite demensional spaces. (3) The construction of the differential geometry on the infinite dimensional spaced such as De Rham decomposition theorem. 2. The research of Markov process : We studied the following problems for Marlkov processes with jumps. (1) The estimate of the fundamental solution. (2) The homogenization theorems. (3) Criteria for the recurrence (transience). 3. The research of the limit theorems : We studied the probability distributions related to limit theorems. In particular, we showed that the distribution of a self similar process with independent increment is Ldistribution and an arbitrary Ldistributio is realized by the above stochastic process. 4. The research of stochastic processes in random environments. We studied the asymptotic behaviors as t of random walks and diffusions in random environment. 5. The research of Gaussian measures and Gaussian processes. As a result, it turned out that the capacity of the channel with feedbacks is not greater than the twice of the capacity of the channel without feedback. 6. The research of probability theory and spectrum. We studied the following objects. (1) The spectrum of the Schrodinger operators in a magnetic field. (2) The spectrum of the Laplacian in Sierpinsky's carpet and gaskets. 7. The research of the ergodic theorems. We studied the structures of Anozov flow, holocycle flow and other dynamical systems. Also, we studied the relation between the local limit theroems related to continued fraction expansions and prime number theorem.
