Quantum white noise and nonlinear infinite dimensional analysis

我们发展了量子白色噪声演算，在其中布朗运动和泊松过程是统一的，并且由于哈德逊和Parthasarathy，推广了量子Ito演算.本研究的主要内容如下：（1）正规序白色噪声微分方程：通过研究算子符号的详细结构，发展了白色噪声算子理论。证明了量子随机微分方程的白色噪声推广的白色噪声微分方程解的存在性、唯一性和正则性。（2）量子白色噪声的高次幂与Levy Laplacian：阐明了量子白色噪声的平方与Levy Laplacian的关系。通过对角化和直接积分构造了一个与Levy Laplacian相关的随机过程。（3）复白色噪声。将Segal-Bargmann变换推广到白色噪声算子，在不假设白色噪声函数空间为核函数的情况下，证明了算子符号的一个新的刻画.得到了白色噪声算子的幺正性准则，并导出了正规序白色噪声方程的解是幺正的充要条件。（4）物理应用：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，并将数学问题公式化。在Fock空间链中观察到Bogoliubov变换的类似物的出现。（5）代数概率论：利用量子分解建立了一种大型图的渐近谱分析的新方法。我们构造了新的例子，并推导出一个必要条件满足的概率措施出现在特定的标度限制。我们研究了这个条件，以获得另一类概率测度，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