This research will study problems related with diffusion processes. These special Gaussian processes, with Markov property, arise mainly as limits of models used in many areas such as microscopic biological phenomena or various computer communications and manufacturing systems. Professor Williams will study diffusion processes in three or more dimensions. The first problem will study reflected Brownian motion in a polyhedral domain. Professor Williams has shown necessary and sufficient conditions for the existence of a two dimensional diffusion with given geometric data. She will now explore these problems in three or more dimensions. Professor Williams plans to obtain characterizations of the stationary distributions and numerical measures of performance for the reflected Brownian motions arising as diffusion approximations to closed queueing networks and to networks with heterogeneous customer populations. Proposed study of the excursions of diffusions in Lipschitz domains will provide important insight to the sample path properties of diffusions. Diffusions with discontinuous/degenerate diffusion coefficients are called singular diffusions. Study of these processes is of great importance to areas such as optimal stopping, filtering and stochastic control.
