Mathematical Sciences: Partial Differential Equations and Probability

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This project will consider regularity of solutions to degenerate elliptic equations, controlled diffusion processes, and Ito's stochastic equations. A variety of techniques from partial differential equations and probability will be utilized. These include barriers, study of boundary operators, and quasiderivatives. The results of this project, in addition to furthering our understanding of the nature of solutions to partial differential equations, will have applications to areas such as finance, population genetics, and control of chemical systems.