2000 Fiscal Year Final Research Report Summary
A study of identification problem for continuous model in phenomena of complex system based on the theory of Langevin equations from the viewpoint of the theory of stochastic processes
| Project/Area Number |
10440026
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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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 | University of Tokyo |
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Principal Investigator |
OKABE Yasunori Graduate School of Engineering, University of Tokyo, Professor, 大学院・工学系研究科, 教授 (30028211)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1998 – 2000
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| Keywords | KM_2O-Langevin equation / KMO-Langevin equation / weak stationarity / purely non-determinicity / fluctuation-dissipation theorem / local and global canonical representation theorem / innovation method / outer matrix function |
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| 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_2O-Langevin equations for degenerate flows in the three directioins of the analysis of local non-linear information space, the analysis of weight transformations and the linear prediction theory. By using these results, we resolved not only the non-linear prediction problem for one-dimensional strictly stationary processes which had remained to be solved for a long time after Masani-Wiener's work, but also both the non-linear prediction problem and the non-linear filtering problem for multi-dimensional stochastic processes with discrete time.
On the other hand, we constructed Kubo noise associated with multi-dimensional stationary flow with continuous time in a Hilbert space.
Next, by taking a procedure of scaling limits of KM_2O-Langevin data that determines the local stochastic difference equation (KM_2O-Langevin
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equation) describing the time evolution of weakly stationary process with discrete time, we derived KMO-Langevin data that determines the global stochastic difference equation (KMO-Langevin equation). Conversely, for a class of weakly stationary process with continuous time, we derived KM_2O-Langevin equation from KMO-Langevin 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 KMO-Langevin data associated with X as a scaling limits with respect to ∈ of KMO-Langevin data associated with X_∈. Less
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