B-值广义白噪声泛函的混沌分解

本文给出了Banach空间值广义白噪声泛函的混沌分解     

白噪声分析中广义算子值函数的Bochner-Wick积分

白噪声广义算子在白噪声分析理论及其应用中起着十分重要的作用. 本文主要讨论了白噪声广义算子值函数的积分及相关问题. 主要工作有: 引入了广义算子值测度的概念, 分别讨论了这种测度在象征和算子p-范数意义下的变差及相互关系; 借助于广义算子的Wick积运算, 引入了广义算子值函数关于广义算子值测度的一种积分---Bochner-Wick积分, 讨论了这种积分的性质, 建立了相应的收敛定理并且展示了其在量子白噪声理论中的应用; 探讨了Bochner-Wick积分的Fubini定理及相关问题.     

算子代数上的广义正定函数

本文研究定义于基本函数空间Ｄ或Ｓ上取值于ｖｏｎＮｅｕｍａｎｎ代数或Ｃ－－代数中的广义函数．证明了每个从局部紧的交换群到Ｃ－代数的范数连续正定函数可以表示为正向量值测度的富里埃变换．也得到了广义正定函数和平移不变厄米正定双线性泛函的一般表示定理．     

广义随机Solow经济增长模型

本文在文[1]的基础上进一步拓广了随机Solow经济增长模型．利用白噪声分析理论建立的广义随机Solow经济增长模型，将随机Solow模型推广到包含广义白噪声泛函及具有非可料扩散系数的情形，并且借助U—泛函方法表明了Picard迭代法在此仍十分有效．     

二阶广义分布参数系统的谱分布

应用泛函分析算子理论的方法研究了Hilbert空间中二阶广义分布参数系统的谱分布问题,利用有界线性算子的广义逆给出了所讨论问题的解及解的构造性表达式。这对研究二阶广义分布参数系统的镇定及渐进稳定性问题都有重要的理论价值。     

广义有理样条的几种构造方法

本文利用Ｔｈｉｅｌｅ倒差分方法、Ｐａｄｅ逼近方法、广义Ｑ．Ｄ．算法及ε－算法等构造了几种广义有理样条函数．此外，通过直接法构造了（ｋ－１，ｋ）－型广义有理样条，给出了它的行列式表示和余项表示并证明了广义有理样条算子的存在性、唯一性、齐次性及连续性．     

平面负位势相关的Schrdinger方程的广义Picard原理

设β是复平面上圆盘Ωα＝｛ｚ｜｜ｚ｜＜ａ｝内的一个零容紧致集．考虑Ωβα＝Ωα＼β上的定常 Ｓｃｈｒｄｉｎｇｅｒ方程（－△＋μ）ｕ＝０,其中位势。μ≤０是 Ｋａｔｏ类Ｒａｄｏｎ测度．方程在广义函数意义下的连续解称为μ－调和函数 将在｛ｚ｜｜ｚ｜＝ａ｝上取极限值０的非负μ－调和函数族记为μＨ.对Ωβα的Ｋｅｒｅｋｊａｔｏ－Ｓｔｏｉｌｏｗ意义下的理想边界β的任一点ζ,本文通过定义μＨ→μＨ的线性算子πζ,引入μＨ的子函数族Ｈζ＝｛ｕ∈μＨ｜πζ（ｕ）＝０｝,证明了在Ωβα上关于ζ的μ－广义Ｐｉｃａｒｄ原理成立,即μＨ的维数是 １或μＨ／Ｈζ的维数是 １二者必居其一．     

Peetre K—泛函和Riesz可和算子

引进了Ｆｏｕｒｉｅｒ－Ｌｅｇｅｎｄｒｅ展开的广义Ｒｉｅｓｚ可和算子。讨论了广义Ｒｉｅｓｚ可和算子的收敛性。建立了广义Ｒｉｅｓｚ可和算子和Ｐｅｅｔｒｅ Ｋ－泛函之间的渐近等价关系。Ｋ－泛函完全刻划了Ｒｉｅｓｚ可和算子的逼近阶。     

广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

广义重调和算子及其在薄板弯曲中的应用

本文用δ－函数具体构造出广义重调和算子，建立相应的二次泛函表达式，并将其应用于弹性薄板的弯曲问题．结果表明．当自变量函数为广义函数时，变分泛函中的自变量函数自然就允许某种程度的不连续性，用Ｌａｇｒａｎｇｅ乘子法所得的修正变分原理实际上是文中给出的变分原理的特殊形式．     

白噪声分析中的梯度算子和散度算子

在白噪声分析的框架中，我们给出了广义Weiner泛函空间上的梯度算子和散度算子的定义与公式，并利用梯度和散度算子以及适应投影建立了广义泛函的表示公式．也证明了积分核算子可用梯度与散度算子表出．     

Generalized Brownian functionals and the Feynman integral

To obtain a sufficiently rich class of nonlinear functionals of white noise, resp. the Wiener process, we study riggings of the L² space with the white noise measure. Particular examples are local functionals such as e.g. the ‘square of white noise’ and its exponential with applications in the theory of Feynman Integral.     

?B-值广义泛函值函数可微性的刻画

In this paper, we discuss fuzzy simplex and fuzzy convex hull, and give several representation theorems for fuzzy simplex and fuzzy convex hull. In addition, by giving a new characterization theorem of fuzzy convex hull, we improve some known results about fuzzy convex hull.     

Generalized Poisson Functionals

Summary With the aim of treating nonlinear systems with inputs being discrete and outputs being generalized functions, generalized Poisson functional are defined and analysed, where the-transforms and the renormalizational play essential roles. For Poisson functionals, the differential operators with respect to a Poisson white noise (t), their adjoint operators and the multiplication operators by (t) are defined. Since these operators involve the time parameter explicitly, they can be used to obtain information concerning the Poisson functional at each point in time. As an example, a new method for measuring the Wiener kernels of such functionals is outlined.     

Intersection Local Times as Generalized White Noise Functionals

For any dimension we present the expansions of Brownian motion self-intersection local times in terms of multiple Wiener integrals. Suitably subtracted, they exist in the sense of generalized white noise functionals; their kernel functions are given in closed (and remarkably simple) form.     

Smoothness of Densities of Generalized Locally Non-Degenerate Wiener Functionals

Several criteria for existence of smooth densities of Wiener functionals are known in the framework of Malliavin calculus. In this article, we introduce the notion of generalized locally non-degenerate Wiener functionals and prove that they possess smooth densities. The result presented here unifies the earlier works by Shigekawa and Florit-Nualart. As an application, we prove that the law of the strong solution to a stochastic differential equation driven by Brownian motion admits a smooth density without an assumption of Lipschitz continuity for dispersion coefficients.     

Analysis of generalized Lévy white noise functionals

Employing the Segal-Bargmann transform (S-transform for abbreviation) of regular Lévy white noise functionals, we define and study the generalized Lévy white noise functionals by means of their functional representations acting on test functionals. The main results generalize (Gaussian) white noise analysis initiated by T. Hida to non-Gaussian cases. Thanks to the closed form of the S-transform of Lévy white noise functionals obtained in our previous paper, we are able to define and study the renormalization of products of Lévy white noises, multiplication operator by Lévy white noises, and the differential operators with respect to a Lévy white noise and their adjoint operators. In the courses of our investigation we also obtain a formula for the products of multiple Lévy-Itô stochastic integrals. As applications, we discuss the existence of Hitsuda-Skorokhod integral for Lévy processes, Kubo-Takenaka formula for Lévy processes, and Itô formula for generalized Lévy white noise functionals.     

A Moment Characterization of B-Valued Generalized Functionals of White Noise

Kernel theorems are established for Bananch space-valued multilinear mappings, A moment characterization theorem for Banach space-valued generalized functionals of white noise is proved by using the above kernel theorems. A necessary and sufficient condition in terms of moments is given for sequences of Banach space-valued generalized functionals of white noise to converge strongly. The integration is also discussed of functions valued in the space of Banach space-valued generalized functionals.     

Implementing Symmetries of a Free Bose Field via White Noise Operators

Linear symmetries of a free Bose field are exploited in the framework of Hida's white noise functionals triple. General symplectic automorphisms on the single particle space are implemented by generalized operators. The intertwining operators are constructed in a physically intuitive way, characterized analytically in terms of symbols, and factorized into three fundamental parts according to Wick ordering procedure. In particular, the classical Shale's theorem is rederived.     

Stochastic Differentiation – A Generalized Approach

The space (D ^*) of Wiener distributions allows a natural Pettis-type stochastic calculus. For a certain class of generalized multiparameter processes X: R ^N (D ^*) we prove several differentiation rules (Itô formulas); these processes can be anticipating. We then apply these rules to some examples of square integrable Wiener functionals and look at the integral versions of the resulting formulas.