Classical and Noncommutative Processes

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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.