矩阵值随机过程与非碰撞扩散粒子系统的对称性
Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
摘要：
作为标准高斯随机矩阵系综Dyson布朗运动模型理论的推广，本文系统地研究了手征和非标准随机矩阵系综的厄米矩阵值过程及其特征值过程.除了非碰撞的布朗运动，我们引入了一个单参数家庭的时间均匀的非碰撞系统的贝塞尔过程和两个参数家庭的时间不均匀的非碰撞系统的Yor的广义蜿蜒和显示，所有的10类特征值统计的Altland-Zirnbauer分类实现为粒子分布在特殊情况下，这些扩散粒子系统。作为时间非齐次特征值过程和非碰撞扩散过程在分布上等价的推论，建立了酉群上积分的Harish-Chandra（Itzykson-Zuber）公式的一个版本的随机微积分证明.
英文摘要：
As an extension of the theory of Dyson's Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor's generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish-Chandra (Itzykson-Zuber) formula of integral over unitary group is established.
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