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，这些理论模拟了具有重大实际意义的物理现象。从随机过程的条件均值和条件方差的直观公式出发，研究与之相关的一类正交鞅多项式，并利用它们来构造新的马尔可夫过程。这些多项式将由q对易方程描述，它将在非对易概率和对易概率之间建立新的联系，并将不同的理论联系在一起，如Hammersley的金属晶体结构长程取向错误的概率模型，以及湍流对流的Frisch和Bourret准弹塑性模型。该项目将促进对经典概率和非对易概率之间联系的理解，经典概率是统计学的基础，非对易概率是量子物理的基础。它将通过个人研究、与研究生的研究以及与美国和欧洲的几位数学家的合作活动来推进。这些成果将通过利用因特网、在会议和讲习班上作陈述以及通过研究出版物广泛和及时地传播。

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.