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

随机分析在随机现象的数学分析中得到了相当大的发展，其传统的框架，称为伊藤演算，是基于布朗运动的泛函。在这个研究项目中，我们从量子白噪声的新方面开始，这是通过对布朗运动应用量子分解和时间导数得到的。由于量子白噪声是更主要的，我们自然会在传统的随机分析之外进行两种研究：非线性理论和非高斯扩展。我们通过研究合作、研讨会、暑期学校和国际会议发展了量子白噪声理论。主要研究内容如下：(1)非线性白噪声方程：利用一个新得到的表征定理，建立了白噪声算子理论，并引入了量子白噪声导数的概念。(2)复白噪声：证明了傅里叶-高斯t…More变换的统一性。我们发现包含二次量子白噪声的正序白噪声方程的解具有部分一致性。(3)利维拉普拉斯和高次量子白噪声：构造相关的随机过程，并通过热方程建立它们之间的关系。(4)相互作用Fock空间：我们建立了一个系统的图的渐近谱分析方法，特别是对于增长正则图和与独立概念相关的图。(5)非高斯随机过程：我们正在继续研究多模相互作用的Fock空间对随机过程的量子分解。(6)无穷维调和分析：应用量子分解的方法证明了Kerov中心极限定理的一个推广。建立了对称群和无限环积的渐近表示理论。(7)量子物理应用：通过量子随机微分方程、湍流统计性质、图上的玻色-爱因斯坦凝聚等促进与物理学家的合作。少

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