Diffusion Processes and Partial Differential Equations

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Diffusion Processes and Partial Differential equations Abstract of Proposed Research Nicolai V KrylovThe project relates to some important modern topics in the theories of Diffusion Processes and Partial Differential Equations (PDEs). They arise from practical applications and include optimal control of random processes, optimal filtering of diffusion processes, white noise driven stochastic PDEs (SPDEs) arising in population genetics, the theory of fully nonlinear PDEs. The project includes an investigation of the numerical methods of finding solutions.Fully nonlinear PDEs arise in the optimal mass transportation problem and in geometry. Rigidity and other characteristics of all kinds of hulls are described in terms of such equations. Control problems and fully nonlinear PDEs also arise in engineering, target tracking, pattern recognition, and many other areas. There are many random processes which we want to control, for instance, the performance of a portfolio or the trajectory of a missile. In target tracking it is important to emphasize that the trajectory is usually only observed with certain errors or noises. Thus the first problem is to filter the noise out of the observations. Such problems were first solved by Kalman and Bucy, who constructed and used their filter during the Apollo program. Currently a major problem is to obtain better results in such problems as weather forecasting, which is a very practical application of SPDEs and the theory of filtering and prediction of random processes."