In their previous work, the proposers have used probabilistic methods to establish an optimal extension of Hormander's theorem for second order differential operators. Their extension asserts hypoellipticity under hypotheses that allow the usual Lie algebra condition to fail at an exponential rate on a collection of hypersurfaces. The proposers use these techniques to investigate a series of further problems in the areas of partial and stochastic differential equations. In particular, they study the Dirichlet problem for exponentially degenerate operators on smooth domains in Euclidean space. They also investigate the existence of smooth densities for a large class of highly degenerate stochastic functional equations. The research deals with two important problem areas that arise in physics and engineering. The first area concerns an important class of mathematical models, called partial differential equations, that play a fundamental role in the study of heat conduction, electric potential and fluid flow. The second area of investigation is devoted to a class of models that are used in physics, engineering and biology in order to analyze dynamical systems whose evolution is influenced by random fluctuations and past history. These models are very important in a variety of diverse areas ranging from signal processing, stock market fluctuations, economic and labor models, aircraft dynamics, materials with memory and population dynamics. The investigators use techniques from the calculus of probability in order to develop a deeper understanding of the above-mentioned models.
