Classical and Noncommutative Processes

这项研究将促进对不同理论之间新的数学联系的理解，这些理论对具有重大实际意义的物理现象进行建模。从随机过程的条件均值和条件方差的直观公式出发，提出者将研究相关联的正交鞅多项式类，并利用它们来构造新的马尔可夫过程。 这些多项式将由q-交换方程描述，该方程将提供非交换概率和交换概率之间的新联系，并将把不同的理论联系在一起，就像哈默斯利的长期-金属晶体结构中的范围取向差以及湍流对流的Frisch和Bourret准随机模型。该项目将促进对经典概率（统计学的基础，以及作为量子物理学基础的非对易概率。 它将通过个人研究，通过与研究生的研究，并通过与美国和欧洲的几位数学家的合作活动来推进。 将通过使用因特网、在会议和讲习班上介绍情况以及通过研究出版物，广泛和及时地传播研究结果。

This research will advance the understanding of new mathematical connections between different theories that model physical phenomena of significant practical interest. Starting from intuitive formulas for the conditional means and conditional variances of stochastic processes, the proposer will study the class of associated orthogonal martingale polynomials and use them to construct new Markov processes. These polynomials will be described by a q-commutation equation which will provide a new link between the noncommutative and commutative probability, and will tie together theories as different as Hammersley's probabilistic models of long-range misorientation in the crystalline structure of metals and Frisch and Bourret parastochastic models of turbulence convection.The project will advance the understanding of connections between the classical probability which is the foundation of statistics, and the non-commutative probability which is a foundation of quantum physics. It will be advanced through personal research, through research with graduate students, and through cooperative activities with several mathematicians in USA and in Europe. The results will be disseminated broadly and in a timely manner through the use of the Internet, through presentations at conferences and workshops, and through research publications.