Quantum Probability Theory and Infinite Dimensional Analysis

我们通过揭示和利用隐藏在背后的非交换结构来研究随机现象的数学模型。引入了量子白噪声导数的新概念，并将其应用于量子随机积分的推广、Fock空间算子的积分表示、量子鞅的表示以及Bogoliubov变换的表征。对于复杂网络的结构分析，我们发展了量子概率方法来分析随机游走的谱分析和转移概率，并研究了自由概率、自由熵和矩阵不等式。

We investigated mathematical models of random phenomena by revealing and utilizing non-commutative structures hidden behind. We introduced the new concept of quantum white noise derivative and applied to generalization of quantum stochastic integrals, integral representation of Fock space operators, representation of quantum martingales, and characterization of Bogoliubov transformation. For the structural analysis of complex networks, we developed the quantum probabilistic approach to spectral analysis and transition probability of random walks, and studied free probability, free entropy, and matrix inequalities.