Complex Stochastic Systems

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The study of stochastic processes is concerned with mathematical descriptions of natural phenomena governed by "random" or "chance" mechanisms. Mathematical models of such phenomena may attempt to describe variation in time, in space, or both. The research to be preformed is concerned with developing methods for specifying these mathematical models, approximating complex models by simpler ones, and constructing models addressing specific scientific applications. This research consists of three parts. The first part explores the theoretical foundations of the martingale characterization as well as applications to nonlinear filtering, and a new approach to understanding the physical phenomenon of metastability. The second part aims to develop new methods of representing stochastic partial differential equations and measure-valued processes in terms of large aggregations of "particles." These particle representations provide powerful tools for analyzing and approximating the associated models. The third part aims to systematically develop stochastic models for chemical reaction networks, beginning with classical Markov chain models and developing new models that take into account the stepwise development of reactions involving RNA and DNA molecules.The proposed research is motivated by interdisciplinary problems and addresses specific applications in physics and chemistry. In addition, the theory and methodology to be developed have applications in other areas such as computer science and finance. The project will also provide a fertile training ground for graduate students and postdoctoral researchers. In particular, there is a high demand for well-trained mathematical scientists with the interest and expertise necessary to contribute to the solution of problems arising in cell and molecular biology.