Mathematical Sciences: Multidimensional Diffusions and Applications

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The investigator is one of the leading authorities in the world on reflected Brownian motions in polyhedral domains. Such processes arise as limits of many practical models, particularly in queuing networks. Investigator's previous results will be extended to three and more dimensions, making the models more applicable. The project will continue to investigate the local path properties of Brownian excursions in Lipschitz domains. In particular, to investigate the distributional properties, it is conjectured that the excursions from tangent hyperplanes will be similar to the excursions from hyperplanes. Diffusions with singular drifts or discontinuous diffusion coefficients arise in many important area of applications such as filtering theory and stochastic control theory. The investigator will explore the questions of existence and uniqueness of such diffusions as well as the return to origin.