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Nonlinear theory of quantum white noise and its non-Gaussian extension

量子白噪声的非线性理论及其非高斯推广

基本信息

批准号：
15340039
负责人：
OBATA Nobuaki
金额：
$ 9.92万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340039/
关键词：
White noise theory Quantum probability Quantum white noise Quantum decomposition Quantum central limit theorem Levy Laplacian Dissipative quantum system Spectral analysis of graphs

项目摘要

Stochastic analysis has developed considerably keeping profound applications to mathematical analysis of random phenomena, where the traditional framework, called the Ito calculus, is based upon functionals of Brownian motion. We start, in this research project, with the new aspect of quantum white noise which is obtained by applying quantum decomposition and time derivative to the Brownian motion. Since the quantum white noise is more primary, we come naturally to two lines of research beyond the traditional stochastic analysis : nonlinear theory and non-Gaussian extension. We developed quantum white noise theory by making research collaboration, workshops, summer schools, international conferences. The research subjects are as follows :(1) Nonlinear white noise equations : We developed white noise operator theory by means of a newly obtained characterization theorem and introduced the notion of quantum white noise derivatives.(2) Complex white noise : Unitarity of the Fourier-Gauss t … More ransform is proved. We found that the solution to normal-ordered white noise equation involving quadratic quantum white noise possesses partial unitarity.(3) Levy Laplacian and higher powers of quantum white noises : Associated stochastic processes are constructed and the relation between them is established through the heat equation.(4) Interacting Fock space : We established a systematic approach to asymptotic spectral analysis of graphs, in particular, for growing regular graphs and graphs related to notions of independence.(5) Non-Gaussian stochastic processes : We are continuing the study of interacting Fock space of multi-mode towards quantum decomposition of stochastic processes.(6) Infinite dimensional harmonic analysis : The method of quantum decomposition is applied to prove a generalization of the Kerov central limit theorem. The asymptotic representation theory of the symmetric groups and the infinite wreath products have been developed.(7) Application to quantum physics : We promoted collaboration with physicists through quantum stochastic differential equations, statistical properties of turbulence, Bose-Einstein condensation on graphs. Less

随机分析在随机现象的数学分析中得到了相当大的发展，其中传统的框架，称为伊藤演算，是基于布朗运动的泛函。在本研究计画中，我们从量子白色杂讯的新观点出发，将量子分解与时间导数应用于布朗运动。由于量子白色噪声是更为主要的噪声，因此，在传统随机分析的基础上，我们自然而然地将研究引向两条主线：非线性理论和非高斯扩展。我们通过研究合作、研讨会、暑期学校、国际会议发展了量子白色噪声理论。（1）非线性白色噪声方程：利用一个新得到的特征定理发展了白色噪声算子理论，并引入了量子白色噪声导数的概念。(2)复白色噪声：傅里叶-高斯变换的么正性 ...更多信息 transform被证明。我们发现含有二次量子白色噪声的正态有序白色噪声方程的解具有部分么正性。(3)Levy Laplacian和量子白色噪声的高次幂：构造了相关的随机过程，并通过热方程建立了它们之间的关系。(4)交互Fock空间：我们建立了一个系统的方法，渐近谱分析的图，特别是，不断增长的定期图和图的概念相关的独立性。(5)非高斯随机过程：我们正在继续研究多模相互作用Fock空间，以实现随机过程的量子分解。(6)无限维调和分析：量子分解的方法是用来证明推广的Kerov中心极限定理。发展了对称群和无限圈积的渐近表示理论。(7)量子物理学应用：我们通过量子随机微分方程，湍流的统计特性，图形上的玻色-爱因斯坦凝聚促进了与物理学家的合作。少