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

为了利用离散时间弱平稳过程来研究连续时间弱平稳过程，我们从局部非线性信息空间分析、权变换分析和线性预测理论三个方向发展了退化流的km_20 - langevin方程理论。利用这些结果，不仅解决了Masani-Wiener之后长期未解决的一维严格平稳过程的非线性预测问题，而且解决了具有离散时间的多维随机过程的非线性预测和非线性滤波问题。另一方面，我们在Hilbert空间中构造了与连续时间多维平稳流相关的Kubo噪声。其次，通过对km_20 - langevin数据的标度限制，得到了描述弱平稳过程离散时间演化的局部随机差分方程（km_20 - langevin…More方程），得到了决定全局随机差分方程（KMO-Langevin方程）的KMO-Langevin数据。反之，对于一类具有连续时间的弱平稳过程，我们利用滤波理论中的创新方法思想，由KMO-Langevin方程推导出km_20 - langevin方程。由此，我们可以从离散特征和局部特征得到离散特征和全局特征的计算算法，并从离散情况下的外矩阵函数表示定理证明连续情况下的外矩阵函数表示定理。因此，我们不仅在局部情况下，而且在全局情况下，完成了描述离散时间弱平稳过程时间演化的随机差分方程的推导。此外，对于一类弱平稳过程X=(X(t)；t∈R)，我们定义对于每个正数∈，我们定义一个随机过程X_∈=(X(n∈)；n∈Z)。然后，我们研究了KMO（代表）。KM_2O)-Langevin数据决定了KMO的耗散项和波动项。描述弱平稳过程X的时间演化的KM_2O -Langevin方程和KMO的确定的∈依赖性。KM_2O)-Langevin数据决定了KMO的耗散项和波动项。KM_<> 0)-Langevin方程，描述弱平稳过程X_∈的时间演化。特别地，我们可以将与X相关的KMO-Langevin数据表示为与X_∈相关的KMO-Langevin数据在∈上的标度极限。少

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 … More 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