Basic Studies on Analysis of Stochastic Variation.

我们在无限维流形上构造了一类与白噪声分析中的Hida微分有关的新型拉普拉斯增量p。这个拉普拉斯算子与著名的Hida拉普拉斯算子Delta_H完全不同。我们的拉普拉斯算子被很好地实现为一个具有C^*不变性的算子，以至于我们可以证明De Rham-Hodge-Kodaira分解定理的无限维版本。我们定义了一种伪傅里叶-迈勒变换(PSI，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变换的缠绕性质和Fock展开。此外，我们还证明了一族PFM变换族是线性同胚群的正则单参数子群，并确定了相应的无穷小生成元。此外，我们还可以将其推广到更一般的变换，并证明了刻划定理。考虑随机变分方程解，得到了CHI(S)在白噪声空间的解，并得到了CHI的显式表示。最后，在白噪声分析的一般背景下，我们考虑了随机边值问题。基于随机渐近解的公式，我们可以构造解。从随机系统振动讨论的观点出发，给出了这类解的一个极限定理。

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.