Multidimensional Diffusions and Applications

9703891 Williams在应用中出现的多维扩散的一些数学问题正在研究中。这些分类在三个标题下。(i)大流量下多类排队网络的逼近。大多数排队网络模型不能精确地分析，必须用更易于处理的模型来近似。一类被称为反射布朗运动的扩散过程，已被证明近似于重负荷的单类和一些多类排队网络。然而，目前还没有一个通用的极限定理来证明这种近似适用于带反馈的高负载多类排队网络。虽然已知这种近似并不具有完全的普遍性，但PI试图确定它们确实适用的多类网络的广泛类别，并提供极限定理来支持这种近似。（ii）反映布朗运动的长期行为。为了获得排队网络模型性能的近似度量，并作为一个内在的数学兴趣问题，考虑反映布朗运动的长期行为是很自然的。特别是，通过研究相关的确定性动力系统，寻求反映布朗运动的半鞅正递推的条件。(3)具有不连续/退化系数的随机微分方程及其在非线性时间序列中的应用。近年来，在时间序列分析中对非线性模型的需求越来越明显。连续时间模型可以极大地方便对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。这些是由具有不连续和退化扩散系数的随机微分方程定义的。当模型的阶数大于2时，解决这些方程解的唯一性问题是今后使用这些模型的一个迫切问题。这个问题，以及它的多变量版本，正在研究中。多维扩散过程作为随机模型广泛应用于物理、生物、工程和社会科学等领域。在这些应用的推动下，人们正在研究许多关于扩散的数学问题。举例来说，排队网络对于分析计算机系统、通信网络和复杂制造系统中的拥塞和延迟具有重要意义。这些网络可以处理不止一类的客户或工作（例如，语音、视频和数据可以通过通信网络中的每个节点），具有复杂的反馈结构（例如，在半导体晶圆制造的可重入线路中），并且经常是重载的。一般来说，这种模型不能被精确地分析，人们自然会寻求易于处理的近似，例如多维扩散。一个具有挑战性的数学问题是这样的近似什么时候是有效的？近年来的研究结果表明，对于带反馈的多类网络来说，这是一个复杂的问题。这个问题和其他多维扩散问题正在这项资助下进行研究。

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.