Mathematical Sciences: Nonlinear Stochastic Models

非线性随机模型，特别是马尔可夫模型或可以用马尔可夫方法分析的非线性随机模型，是本文的研究重点。其中包括时间序列分析中非常感兴趣的非线性自回归（移动平均）过程，以及随着参数变化而表现出各种相变的动力系统和流动的随机扰动。本文探讨的马尔可夫过程和动力系统之间的相互作用对这两个领域都有重要的影响。对于那些随着时间的推移趋于唯一稳态的马尔可夫过程，进一步的研究对象是中心极限定理及其渐近展开式的细化。这些改进在不同的领域有应用，如时间序列的自举，一类偏微分方程的大时间解的奇异摄动展开，这些方程在预测含水层中污染物的扩散时出现。自然界中发生的随机过程通常可以看作是受随机噪声干扰的确定性系统。正如确定性部分的性质极大地影响随机过程的行为一样，随机过程的统计行为也可用于预测确定性系统本身的大时间行为。后一项研究在分析复杂的确定性系统时特别有用，例如在湍流研究中出现的系统。目前的建议旨在探索确定性和随机过程之间的相互作用。另一个目标是细化控制许多重要随机现象的长期平均行为的精确统计规律。在各种潜在的应用中包括时间序列的自举和含水层中污染物扩散的预测。

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.