Project/Area Number 
10440026

Research Category 
GrantinAid for Scientific Research (B).

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  University of Tokyo 
Principal Investigator 
OKABE Yasunori Graduate School of Engineering, University of Tokyo, Professor, 大学院・工学系研究科, 教授 (30028211)

CoInvestigator(Kenkyūbuntansha) 
INOUE Akihiko Graduate School of Science, Hokkaido University, Associate Professor, 大学院・理学研究科, 助教授 (50168431)
AIHARA Kazuyuki Graduate School of Frontier Sciences, University of Tokyo, Professor, 大学院・新領域創成科学研究科, 教授 (40167218)
YANAGAWA Takashi Graduate School of Science, Kyushu University, Professor, 大学院・数理学研究院, 教授 (80029488)
MATSUURA Masaya Graduate School of Engineering, Assistant Professor, 大学院・工学系研究科, 助手
HORITA Takehiko Graduate School of Engineerin, Lecturer, 大学院・工学系研究科, 講師 (90222281)
伏見 正則 東京大学, 大学院・工学系研究科, 教授 (70008639)

Project Period (FY) 
1998 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥6,700,000 (Direct Cost: ¥6,700,000)
Fiscal Year 2000: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1999: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1998: ¥2,400,000 (Direct Cost: ¥2,400,000)

Keywords  KM_2OLangevin equation / KMOLangevin equation / weak stationarity / purely nondeterminicity / fluctuationdissipation theorem / local and global canonical representation theorem / innovation method / outer matrix function / 純非決定性,完全ランク性とテープリッツ条件 / 連続系と離散系 / 多次元非線形予測解析 / KMOランジュヴァンデータ / KM_2Oランジュヴァンデータ / スケール極限 / 純非決定性完全ランク性とテープリッツ条件 
Research Abstract 
As a preparation for the aim to investigate weakly stationary process with continuous time by using weakly stationary process with discrete time, we developed the theory of KM_2OLangevin equations for degenerate flows in the three directioins of the analysis of local nonlinear information space, the analysis of weight transformations and the linear prediction theory. By using these results, we resolved not only the nonlinear prediction problem for onedimensional strictly stationary processes which had remained to be solved for a long time after MasaniWiener's work, but also both the nonlinear prediction problem and the nonlinear filtering problem for multidimensional stochastic processes with discrete time. On the other hand, we constructed Kubo noise associated with multidimensional stationary flow with continuous time in a Hilbert space. Next, by taking a procedure of scaling limits of KM_2OLangevin data that determines the local stochastic difference equation (KM_2OLangevin
… More
equation) describing the time evolution of weakly stationary process with discrete time, we derived KMOLangevin data that determines the global stochastic difference equation (KMOLangevin equation). Conversely, for a class of weakly stationary process with continuous time, we derived KM_2OLangevin equation from KMOLangevin equation, by using the idea of innovation method in the filtering theory. Thus, we could obtain the algorithm calculating the discrete and global characteristics from the discrete and local characteristics and prove the representaion theorem of outer matrix function for continuous case from the one for discrete case. Therefore, we have completed the derivation of the stochastic difference equation describing the time evolution of weakly stationary process with discrete time not only for local case, but also for global case. Moreover, for a class of weakly stationary process X=(X(t) ; t∈R), we define for each positive number ∈, we define a stochastic process X_∈=(X(n∈) ; n∈Z). Then, we investigated KMO(resp.KM_2O)Langevin data that determines the dissipation term and the fluctuation term in KMO(resp.KM_2O)Langevin equation describing the time evolution of the weakly stationary process X and certatin ∈dependence of KMO(resp.KM_2O)Langevin data that determines the dissipation term and the fluctuation term in KMO (resp.KM_<>O)Langevin equation describing the time evolution of the weakly stationary process X_∈. In particular, we could represent KMOLangevin data associated with X as a scaling limits with respect to ∈ of KMOLangevin data associated with X_∈. Less
