Nonlinear stochastic models, especially those which are either Markovian or may be analyzed by Markovian methods, are the main focus of the proposal. Among these are nonlinear autoregressive (-moving average) processes of much interest in time series analysis, and random perturbations of dynamical systems and flows which exhibit various phase transitions as a parameter changes. The interplay between Markov processes and dynamical systems that this proposal explores has significant consequences to both fields. For those Markov processes which approach a unique steady state as time progresses, a further object of study is the central limit theorem and its refinement by asymptotic expansions. These refinements have applications in diverse areas such as bootstrapping in time series, and singular perturbation expansions of large time solutions of a class of partial differential equations which arise in the prediction of the spread of contaminants in an aquifer. Random processes which occur in nature may often be regarded as a deterministic system perturbed by random noise. Just as the nature of the deterministic part greatly influences the behavior of the random process, the statistical behavior of the random process may also be used in predicting the large time behavior of the deterministic system itself. This latter study is particularly useful in analyzing complex deterministic systems such as those arising in the study of turbulence. The present proposal seeks to explore this interplay between deterministic and random processes. Another object is the refinement of precise statistical laws governing the long-run average behavior of many important random phenomena. Among diverse potential applications are bootstrapping in time series and the prediction of the spread of contaminants in an aquifer.
