拟可加测度空间上广义Sugeno积分的收敛性

在拟可加测度空间上通过引入拟乘算子重新定义广义Sugeno积分,针对依拟可加测度收敛的函数列,应用诱导算子和拟乘算子的运算性质讨论和分析广义Sugeno积分的收敛性,进而获得了这种广义Sugeno积分的单调增收敛定理.     

σ—弱算子拓扑下的测度与积分

我们引入了算子测度和算子值函数的σ-弱积分;证明了Ba（R）等距同构于L（B（X,R）;M）;给出了σ-弱算子拓扑下的Riesz表示定理;并将任一自伴算子表示成某一算子值函数的σ-弱积分。     

有界闭凸值测度的集值积分

引入实值函数关于有界闭凸值测度的集值积分,并讨论了集值积分的收敛定理,证明了当集值测度为有界闭凸集值的有界变差集值测度时,关于弱紧凸集值测度的积分性质对有界闭凸集值测度仍然保持.推广了实值函数关于弱紧凸值测度的积分.     

Fuzzy数测度与积分

本文利用文[2]所给出的Fuzzy数测度的概念,定义了(—)fuzzy值函数关于(—)fuzzy数测度的积分,并且研究了这种积分的性质,得到了各种收敛定理,其中包括广义Lebesgue单调收敛定理、Fatou引理及Lebesgue控制收敛定理。在最后,讨论了(—)fuzzy数测度的R—N导数的存在性,并且给出了Fubini定理。     

复值函数的广义复模糊积分北大核心

本文给出了非负复值函数关于模糊复测度的广义复模糊积分的几种等价形式,并讨论了由该积分表示的几种复模糊积分方程有解的充要条件.在此基础上,进一步引进了一般复值函数的广义复模糊积分的概念,给出了该积分的一些基本性质,并在一定条件下证明了单调收敛定理.     

复值函数的广义复模糊积分

本文给出了非负复值函数关于模糊复测度的广义复模糊积分的几种等价形式,并讨论了由该积分表示的几种复模糊积分方程有解的充要条件.在此基础上,进一步引进了一般复值函数的广义复模糊积分的概念,给出了该积分的一些基本性质,并在一定条件下证明了单调收敛定理.     

K-拟可加模糊数值积分及其收敛性

在K-拟可加模糊测度空间上,针对一类(?)-可积模糊数值函数,建立了所谓的K-拟可加模糊数值积分,并通过引入诱导算子K,获得这种积分的转换定理,进而研究这种K-拟可加模糊数值积分的一些重要性质,同时给出了它的一系列收敛定理,从而丰富了模糊数学的积分理论。     

模糊值函数的t-余模积分

给出了模糊值函数关于t-余模、 -分解测度的t-余模、 -积分(简记为 -积分)的定义,并讨论了模糊值函数 -积分的一些性质和单调收敛定理.这种积分是模糊值函数Lebesgue积分的推广,也是实值函数 -积分的推广.     

广义（N）-模糊积分的转换与表示定理

针对模糊测度空间上由一般可测函数所定义的广义(N)-模糊积分,结合该模糊积分与Lebesgue积分的内在联系,利用α-截断函数的定义,分别首次获得这种广义(N)-模糊积分的积分转换定理和表示定理.     

B -值白噪声广义泛函的解析刻画

Banach空间值白噪声广义泛函是一类重要的向量值白噪声广义泛函. 该文建立了Banach空间值白噪声广义泛函的一个解析刻画定理, 并给出了此结果的若干应用.     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

A Generalized Analytic Operator-Valued Function Space Integral and a Related Integral Equation

We introduce a generalized Wiener measure associated with a Gaussian Markov process and define a generalized analytic operator-valued function space integral as a bounded linear operator from L _p into L _p^\prime (1<p ≤ 2) by the analytic continuation of the generalized Wiener integral. We prove the existence of the integral for certain functionals which involve some Borel measures. Also we show that the generalized analytic operator-valued function space integral satisfies an integral equation related to the generalized Schr?dinger equation. The resulting theorems extend the theory of operator-valued function space integrals substantially and previous theorems about these integrals are generalized by our results.     

A Generalized Analytic Operator-Valued Function Space Integral and a Related Integral Equation

We introduce a generalized Wiener measure associated with a Gaussian Markov process and define a generalized analytic operator-valued function space integral as a bounded linear operator from L _p into L _p^\prime (1<p ≤ 2) by the analytic continuation of the generalized Wiener integral. We prove the existence of the integral for certain functionals which involve some Borel measures. Also we show that the generalized analytic operator-valued function space integral satisfies an integral equation related to the generalized Schr?dinger equation. The resulting theorems extend the theory of operator-valued function space integrals substantially and previous theorems about these integrals are generalized by our results.     

Unitarity of Generalized Fourier–Gauss Transforms

Abstract

A generalized Fourier–Gauss transform is an operator acting in a Boson Fock space and is formulated as a continuous linear operator acting on the space of test white noise functions. It does not admit, in general, a unitary extension with respect to the norm of the Boson Fock space induced from the Gaussian measure with variance 1 but is extended to a unitary isomorphism if the Gaussian measure is replaced with the ones with different covariance operators. As an application, unitarity of a generalized dilation is discussed.     

The mass-transfer vector problem

The mass-transference problem is studied for Banach valued cost functions and operator-valued measures. The solvability of the primal problem is stated under certain natural conditions, for general measurable functions and measures of bounded variation. The continuous case is also studied, and the solvability and the absence of a duality gap are established for continuous vector functions and regular operator valued measures.     

White Noise of Poisson Random Measures

We develop a white noise theory for Poisson random measures associated with a pure jump Lévy process. The starting point of this theory is the chaos expansion of Itô. We use this to construct the white noise of a Poisson random measure, which takes values in a certain distribution space. Then we show, how a Skorohod/Itô integral for point processes can be represented by a Bochner integral in terms of white noise of the random measure and a Wick product. Further, based on these concepts we derive a generalized Clark–Haussmann–Ocone theorem with respect to a combination of Gaussian noise and pure jump Lévy noise. We apply this theorem to obtain an explicit formula for partial observation minimal variance portfolios in financial markets, driven by Lévy processes. As an example we compute the closest hedge to a binary option.     

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.     

m-Dissipativity for Kolmogorov Operator of a Fractional Burgers Equation with Space-time White Noise

This paper is concerned with the essential m-dissipativity of the Kolmogorov operator associated with a fractional stochastic Burgers equation with space-time white noise. Some estimates on the solution and its moments with respect to the invariant measure are given. Moreover we also study the smoothing properties of the transition semigroup and the corresponding fractional Ornstein-Uhlenbeck operator by introducing an auxiliary semigroup and (generalized) Bismut-Elworthy formula. From these results, we prove that the Kolmogorov operator of the problem is m-dissipative and the domain of the infinitesimal generator of the fractional Ornstein-Uhlenbeck operator is a core.     

A hyperfinite flat integral for generalized random fields

In this paper, a nonstandard construction of generalized white noise is established. This provides a (hyperfinite) flat integral representation of probability measures for generalized random fields derived as image probability measures of generalized white noise under certain measurable transformations, including Euclidean random fields obtained as convolution from generalized white noise with Euclidean kernels.