Mathematical Sciences: Nonlinear Stochastic Models

非线性随机模型，特别是那些 马尔可夫或可以通过马尔可夫方法分析，是 提案的重点。 其中，非线性 自回归（-移动平均）过程， 时间序列分析和动态随机扰动 系统和流动表现出各种相变， 参数变化。 马尔可夫过程和 该提案所探讨的动力系统具有重要意义， 这两个领域的后果。 对于那些马尔可夫过程， 随着时间的推移，接近一个独特的稳定状态， 研究的对象是中心极限定理及其改进 渐近展开。 这些改进可应用于 不同的领域，如时间序列中的自举，和奇异 一类非线性微分方程大时间解的摄动展开 偏微分方程的预测中出现的 污染物在含水层中的扩散。 自然界中发生的随机过程常常可以看作是 被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了行为 随机过程的统计行为 过程也可以用于预测的大时间行为 确定性系统本身。 后一项研究是 特别适用于分析复杂的确定性系统 例如在湍流研究中产生的那些。 本 一项提案试图探索确定性与非确定性之间的相互作用， 随机过程。 另一个目标是改进 长期平均行为的精确统计规律 许多重要的随机现象。 在各种潜力中 应用程序在时间序列和预测中自举 含水层中污染物的传播。

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.