论文标题
对Stone定理的量子随机扩展的调查
Survey on a quantum stochastic extension of Stone's Theorem
论文作者
论文摘要
从Kümmerer对固定马尔可夫流程的调查中，出现了一个对白色噪声的操作员代数定义，该定义捕获了许多示例，以及从经典的概率和非交通概率中。在与白噪声相关的非交换性$ l^p $ - 空间中，（非）交换莱维过程的作用是由添加剂cocycles用于白噪声转移的，而且，经典lévy流程的指数的概念是由无单位的同伴普遍化的。作为主要结果，我们报告了白噪声转移的添加剂和单一共生之间的徒点对应关系。如果需要有差异性，则提出的对应关系减少到Stone定理（对于规范连续统一组）。对应关系需要开发与$ l^\ infty $结合的协方差操作员的添加剂结果：操作员值的随机ITô集成，二次变化和非交通性的Martingale不平等现象以及随机差异。报道了与无限差异操作员的添加剂合过程对添加剂的相关结果和最新进展。
From Kümmerer's investigations on stationary Markov processes has emerged an operator algebraic definition of white noises which captures many examples from classical as well as from non-commutative probability. Within non-commutative $L^p$-spaces associated to a white noise, the role of (non-)commutative Lévy processes is played by additive cocycles for the white noise shift, and moreover, the notion for exponentials of classical Lévy processes is generalized by unitary cocycles. As a main result we report a bijective correspondence between additive and unitary cocycles for white noise shifts. If the cocycles are required to be differentiable, the presented correspondence reduces to Stone's theorem (for norm continuous unitary groups). The correspondence needs the development of background results for additive cocycles with $L^\infty$-bounded covariance operators: an operator-valued stochastic Itô integration, quadratic variations and non-commutative martingale inequalities as well as stochastic differentiation. Related results and recent progress towards the case of additive cocycles with unbounded variance operators are reported.
