Mathematical Sciences: Presidential Young Investigator Award

本研究将研究与扩散相关的问题 流程. 这些特殊的高斯过程， 财产，主要是作为在许多领域使用的模型的限制而出现的 例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 将研究三维或三维以上的扩散过程。 的 第一个问题将研究反射布朗运动， 多面体域 威廉姆斯教授已经证明， 存在二维的充分条件 给定几何数据的扩散。 她现在将探索这些 三个或更多维度的问题。 威廉姆斯教授计划 以获得平稳分布的特征， 反射布朗函数性能的数值测量 作为对封闭扩散的扩散近似的运动 网络以及具有异构客户群体的网络。 Lipschitz中扩散漂移的研究 域将为示例路径提供重要的洞察力 扩散的性质。 扩散， 不连续/退化扩散系数称为 奇异扩散 对这些过程的研究具有重要意义 重要性的领域，如最佳停止，过滤和 随机控制

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.