Mathematical Sciences: Presidential Young Investigator Award

本研究将研究扩散过程的相关问题。这些具有马尔科夫性质的特殊高斯过程主要是由于在许多领域中使用的模型的限制而产生的，如微观生物现象或各种计算机通信和制造系统。威廉姆斯教授将研究三维或更多维度的扩散过程。第一个问题将研究多面体区域中的反射布朗运动。Williams教授在给定几何参数的情况下，证明了二维扩散存在的充要条件。她现在将从三个或更多的维度探讨这些问题。威廉姆斯教授计划获得平稳分布的特征和反射布朗运动的性能的数值度量，这些反射布朗运动作为闭合排队网络和具有不同客户群的网络的扩散近似而产生。对Lipschitz域中扩散的漂移的研究将为研究扩散的样本路径性质提供重要的启示。具有不连续/退化扩散系数的扩散称为奇异扩散。对这些过程的研究在最优停止、滤波和随机控制等领域具有重要意义。

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.