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

随机分析已仔细地在随机现象的数学分析中仔细地应用了，其中传统框架称为ITO微积分，基于布朗运动的功能。我们从这个研究项目开始，从量子白噪声的新方面开始，这是通过将量子分解和时间导数应用于布朗运动而获得的。由于量子白噪声更为主要，因此除了传统的随机分析之外，我们自然而然地进行了两条研究：非线性理论和非高斯扩展。我们通过进行研究合作，研讨会，暑期学校，国际会议来开发量子白噪声理论。研究对象如下：（1）非线性白噪声方程：我们通过新获得的表征理论开发了白噪声操作者理论，并介绍了量子白噪声衍生物的概念。（2）复杂的白噪声：傅立叶 - 高斯的单位性。我们发现，涉及二次量子白噪声电位的正常白噪声方程的解决方案部分。（3）征费的laplacian和较高的量子白噪声的力量：构建相关的随机过程，它们之间的关系是通过热量方程来建立的。（5）非高斯随机过程：我们正在继续研究多模式与随机过程量子分解的相互作用空间。（6）无限尺寸谐波分析：量子分解的方法用于证明Kerov集中限制的普遍性。已经开发了对称基团和无限炉灶产物的不对称表示理论。（7）应用于量子物理学：我们通过量子随机微分方程，湍流的统计特性，Bose-Inerstein Condensation促进了与物理学家的合作。较少的