Diffusion Processes and Partial Differential Equations

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This project focuses on some of the central topics in the modern theory of partial differential equations (PDE) and diffusion processes. The problems arise from practical applications and, in mathematical terms, are described as optimal control of random processes, optimal filtering of diffusion processes, and white-noise-driven stochastic PDE (SPDE). The problems are formulated in the language of the theory of fully nonlinear PDE, and the project includes investigation of numerical methods for finding their solutions.Fully nonlinear partial differential equations arise in a multitude of contexts, including control theory, optimal mass transportation problems, and geometry, to name just three. Rigidity and other characteristics of all kinds of hulls (of, say, ships or missiles) are described in terms of such equations. Control problems and fully nonlinear equations also turn up in engineering, target tracking, pattern recognition, and a host of other applied areas. There are many random processes that it is both desirable and important to control (e.g., the performance of a stock portfolio, the trajectory of a missile). In target tracking, for instance, it is important to emphasize that the trajectory of a projectile is observed, in general, with certain errors or noises. Therefore, the first problem in controlling the trajectory is to filter the noise out of the observations. Such problems were initially solved by Kalman and Bucy, who constructed and used their filter during the Apollo program. Needless to say, much more work needs to be done in order to achieve more accurate results. Improved filters would be relevant to endeavors such as weather forecasting (or, more generally, climate change forecasting), which is one of the possible concrete applications of stochastic partial differential equations and the theory of filtering and prediction of random processes.