Quantum white noise and nonlinear infinite dimensional analysis

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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