Quantum white noise and nonlinear infinite dimensional analysis

我们发展了统一布朗运动和泊松过程的量子白噪声演算，并在Hudson和Parthasarathy的基础上扩展了量子伊托演算。(1)正序白噪声微分方程：通过研究算子符号的详细结构，我们发展了白噪声算子理论。由此证明了一个白噪声微分方程解的存在唯一性和正则性，该方程是量子随机微分方程的白噪声扩展。(2)量子白噪声的高次幂与Levy Laplacian：阐明了量子白噪声的平方与Levy Laplacian的关系。利用对角化和直接积分的方法构造了一个与Levy拉普拉斯算子相关的随机过程(3)复白噪声。将Segal-Bargmann变换推广到白噪声算子，证明了一种新的算子符号表征，而不需要假设白噪声函数空间的核性。(4)在物理上的应用：研究了描述量子耗散模型的各种量子随机微分方程之间的关系，并提出了数学问题。(5)代数概率论：利用量子分解建立了一种大图渐近谱分析的新方法。我们构造了新的例子，并推导了在特定尺度极限下出现的概率测度所满足的必要条件。我们研究这个条件是为了得到另一类概率测度，如q-高斯分布。此外，我们还证明了另一类极限定理

We developed quantum white noise calculus within which Brownian motion and Poisson process are unified and extended quantum Ito calculus due to Hudson and Parthasarathy. Below are the main subjects of this research project(1) Normal-ordered white noise differential equations: We developed white noise operator theory by examining detailed structure of the operator symbols. We thereby proved existence, uniqueness and regularity of a solution to a white noise differential equation which is a white noise extension of a quantum stochastic differential equation(2) Higher powers of quantum white noises and the Levy Laplacian: We clarified a relation between the square of quantum white noises and the Levy Laplacian. A stochastic process associated with the Levy Laplacian was constructed by means of diagonalization and a direct integral(3) Complex white noise. Extending the Segal-Bargmann transform to a white noise operator, we proved a new characterization of operator symbols without assuming the nuclearity of the space of white noise functions. We obtained a unitarity criterion for a white noise operator and derived a necessary and sufficient condition for a solution to a normal-ordered white noise equation to be unitary(4) Application to physics: We studied relation among verious kinds of quantum stochastic differential equations which describe models of quantum dissipation and formulated mathematical problems. We observed emergence of an analogue of Bogoliubov transformation in a chain of Fock spaces(5) Algebraic probability theory: We established a new method of asymptotic spectral analysis of a large graph by means of quantum decomposition. We constructed new examples and derived a necessary condition satisfied by the probability measures emerging in particular scaling limits. We studied this condition to obtain another class of probability measures such as q-Gaussian distributions. Moreover, we proved another type of limit theorems