The first part of this project focuses on infinite systems of stochastic equations. Such systems provide natural models in their own right, but results in this part of the project will also provide essential tools for the study of the stochastic partial differential equations to be considered in the second part. The second part of the project will develop methods for representing solutions of stochastic partial differential equation in terms of infinite systems of "particles." These particle representations provide powerful tools for analyzing and approximating the associated models. In some cases, the particle representations arise naturally from the phenomenological interpretation of the stochastic partial differential equation. In others, they simply provide a mathematical tool for its analysis. The particle systems are typically specified as solutions of infinite systems of stochastic differential equations, and uniqueness results to be developed in the first part of the project will play a critical role in obtaining the desired representations. The third part of the project continues work on models of intracellular reaction networks. One goal will be to provide algorithms that simplify the application of earlier work on model reduction. A second will be to extend the earlier results to models that take into account the spatial structure of the cell.
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 performed is concerned with developing methods for specifying these mathematical models, approximating complex models by simpler ones, and constructing models addressing specific scientific applications. One fundamental approach to specifying models of stochastic phenomena is to formulate equations with stochastic inputs whose solution is the desired model. Different stochastic equations may determine the same model, and different equations may provide different insight into the model and the phenomenon the model addresses. The project is concerned with how to develop and exploit multiple representations for classes of models that arise naturally in physical, biological and financial systems.