Multidimensional Diffusions and Applications

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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.