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Basic Studies on Analysis of Stochastic Variation.

随机变异分析的基础研究。

基本信息

批准号：
07640280
负责人：
DOKU Isamu
金额：
$ 0.96万
依托单位：
Saitama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1997
项目状态：
已结题

项目摘要

We construct on an infinite dimensional manifold a new type of Laplacian DELTA_p associated with the Hida differential in white noise analysis. This Laplacian is completely different from the well known Hida Laplacian DELTA_H. Our Laplacian is realized so nicely as an operator having C^* invariance that we can show an infinite dimensional version of the De Rham-Hodge-Kodaira decomposition theorem. We define a Pseudo-Fourier-Mehler transfrom PSI (PFM transform for short) and derive several fundamental properties. Actually the PFM transform is a variant of infinite dimensional Fourier type transform in white noise analysis. Intertwining properties and Fock expansion of the PFM transfrom are derived. Moreover, we can show that a family of PFM transforms forms a regular one-parameter subgroup of the linear homeomorphism group, and the corresponding infinitesimal generator is determined. In addition, we can extend it to a more generalized transform and the characterization theorem is proved. We consider the stochastic variational equation and find the solution CHI (s) lying in the white noise space, and an explicit representation of deltaCHI is also obtained. Finally, we consider the stochastic boundary value problem under the general setting of white noise analysis. Based on the formulation of stochastic asymptotic solutions, we can construct the solutions. A limit theorem for such solutions is derived from the point of view of oscillatory discussion for the random system.

我们在无限维流形上构造了一种与白噪声分析中的Hida微分相关的新型拉普拉斯DELTA_p。这个拉普拉斯函数与著名的Hida拉普拉斯函数DELTA_H完全不同。我们的拉普拉斯算子被很好地实现为一个C^*不变性的算子我们可以展示一个无限维的De Rham-Hodge-Kodaira分解定理。我们从PSI定义了伪傅立叶-米勒变换（简称PFM变换），并推导了几个基本性质。实际上，PFM变换是白噪声分析中无限维傅里叶变换的一种变体。推导了PFM变换的缠结特性和Fock展开。此外，我们还证明了PFM变换族构成线性同胚群的正则单参数子群，并确定了相应的无穷小发生器。此外，我们可以将其推广到一个更广义的变换，并证明了表征定理。我们考虑随机变分方程，在白噪声空间中找到了CHI (s)的解，并得到了deltaCHI的显式表示。最后，我们考虑了白噪声分析一般情况下的随机边值问题。基于随机渐近解的公式，我们可以构造解。从随机系统的振荡讨论的角度，导出了这类解的极限定理。