Multidimensional Diffusions and Applications

威廉姆斯9703891 中出现的多维扩散的一些数学问题 应用正在研究中。这些问题分为三个标题。(i)近似 在繁忙的交通中的多类路由网络。大多数嵌入式网络模型不能 必须精确分析，必须用更易处理的模型近似。某一类 扩散过程，被称为反射布朗运动，已被证明， 近似重载的单类和一些多类网络。然而，在这方面， 目前还没有一般的极限定理来证明这种近似， 加载了带有反馈的多类神经网络。尽管已知这样的 近似值并不具有完全的普遍性，PI试图确定广泛的类别 它们确实适用的多类网络的性质，并提供极限定理来支持 这种近似。(ii)反射布朗运动的长期行为。为 获得性能的近似度量的目的，用于验证网络模型 作为一个内在的数学兴趣问题，从长远来看， 反射布朗运动的行为。特别是，正递归的条件 的半鞅反映布朗运动正在寻求通过研究一个相关的 确定性动力系统(iii)随机微分方程 不连续/退化系数和非线性时间序列的应用。需要 近年来，时间序列分析中的非线性模型变得越来越明显 年连续时间模型可以极大地促进不规则间隔数据的分析。 一类有用的非线性连续时间模型是连续时间阈值 自回归模型这些是由随机微分方程定义的， 不连续和退化扩散系数。未来使用的一个紧迫问题是， 这些模型是为了解决这些方程的解的唯一性问题， 模型的阶数大于2。这个问题，以及多变量版本的 它，正在研究中。 多维扩散过程被广泛地用作随机模型， 在物理、生物、工程和社会科学中的各种应用。 受这些应用的启发，本文提出了一些扩散的数学问题。 被研究。作为一个说明性的例子，互联网是当前的相关性， 分析计算机系统、通信网络和 复杂的制造系统。这些网络可以处理多于一类的 客户或工作（例如，语音、视频和数据可以通过 通信网络），具有复杂的反馈结构（例如，在可重入线中， 半导体晶片制造）并且经常是重负载的。通常这样 模型不能被精确地分析，人们自然会寻求易于处理的模型。 近似，例如多维扩散。一个具有挑战性的数学问题 这是什么时候这种近似是有效的？近年来的研究结果表明， 对于有反馈的多类网络来说是一个复杂的问题。这个问题和其他问题， 在这项赠款下，正在进行多方面的传播。

9703891 Williams A number of mathematical problems for multidimensional diffusions arising in applications are being studied. These are grouped under three headings. (i) Approximation of multiclass queueing networks in heavy traffic. Most queueing network models cannot be analyzed exactly and must be approximated by more tractable models. A certain class of diffusion processes, known as reflecting Brownian motions, have been shown to approximate heavily loaded single class and some multiclass queueing networks. However, currently there is no general limit theorem to justify such approximations for heavily loaded multiclass queueing networks with feedback. Although it is known that such approximations do not hold in complete generality, the PI seeks to identify broad classes of multiclass networks for which they do hold and to provide limit theorems to support such approximations. (ii) Long run behavior of reflecting Brownian motions. For the purpose of obtaining approximate measures of performance for queueing network models and as a problem of intrinsic mathematical interest, it is natural to consider the long run behavior of reflecting Brownian motions. In particular, conditions for positive recurrence of semimartingale reflecting Brownian motions are being sought by study of a related deterministic dynamical system. (iii) Stochastic differential equations with discontinuous/degenerate coefficients and applications to nonlinear time series. The need for nonlinear models in time series analysis has become increasingly apparent in recent years. Continuous time models can greatly facilitate the analysis of irregularly spaced data. A useful class of nonlinear continuous time models are the continuous time threshold autoregressive models. These are defined by stochastic differential equations with a discontinuous and degenerate diffusion coefficient. A compelling problem for future use of these models is to resolve the problem of uniqueness for solutio ns of these equations when the order of the model is larger than two. This problem, as well as multivariate versions of it, are being studied. Multidimensional diffusions processes are used as stochastic models in a wide variety of applications in the physical, biological, engineering and social sciences. Motivated by such applications, a number of mathematical problems for diffusions are being studied. As an illustrative example, queueing networks are of current relevance for analyzing congestion and delay in computer systems, communication networks and complex manufacturing systems. These networks may process more than one class of customer or job (e.g., voice, video and data may pass through each node in a communication network), have complex feedback structures (e.g., as in reentrant lines for semiconductor wafer fabrication) and are frequently heavily loaded. In general such models cannot be analyzed exactly and one is naturally led to seek tractable approximations, such as multidimensional diffusions. A challenging mathematical question here is when is such an approximation valid? Results of recent years have revealed this to be a complex issue for multiclass networks with feedback. This question and others for multidimensional diffusions are being pursued under this grant.