Finite and Infinite-Dimensional Stochastic Dynamical Systems

本计画主要研究随机微分方程的动力学与数值问题。有限维流形上的随机常微分方程产生流形上的随机流。研究的一个目标是在驱动向量场的适当的正则性和增长条件下，为这种流动在定常解附近构造不变流形。特别是，研究者将证明存在稳定，不稳定和中心流形上的每个驻点附近。为了使sodes构成可行的物理模型，研究者证明了一个Kupka-Smale型定理必须成立。Hilbert空间上一类重要的无穷维半流是由光滑紧致流形上的耗散半线性随机偏微分方程生成的。对于这些半流，研究者打算证明在驻点附近存在有限维不变流形。重要的例子包括本分析的spdes伯格方程，仿射线性随机发展方程和随机反应扩散方程。研究的结果将揭示这些模型的随机动力学的新特征。在许多工程和物理应用中，人们会遇到带记忆随机系统模型。这种模型的解决方案的确定性光滑约束自然导致sfdes（紧）黎曼流形。研究者将研究由流形上的随机微分方程的轨迹生成的路径空间值马尔可夫过程。研究人员将建立一个伊藤公式的无限维段过程，并试图获得其无穷小生成元的几何不变的环境黎曼流形。从路径动力学的观点出发，研究者将在紧黎曼流形上的sfdes的解诱导的路径空间上构造完美半流。该项目的重点是定性和长期行为的一大类概率模型称为随机微分方程。这些方程被科学家和工程师广泛使用。特别感兴趣的是一类用于物理学，工程学和生物学的模型，以分析其演化受随机波动和过去历史影响的动力系统。这些模型在各种不同的领域都非常重要，如信号处理，股票市场波动，经济和劳动力模型，飞机动力学，记忆材料，人口动力学和流体流动。研究人员将使用最新的概率技术，以更深入地了解这些模型。该项目的研究成果将产生关于接近统计平衡状态的基本随机方程解的长期行为的精确信息。数值算法将被开发，从而真实的市场数据将被用来测试期权定价模型，其中股票价格是由其过去的历史。在不同的方向，研究将巩固与现代数学研究的其他领域，特别是动力系统和几何理论的重要联系。该项目的结果将被汇编成一本研究专著，对象是数学、工程和金融专业的研究生。

The research in this project focuses on several dynamical and numerical aspects of stochastic differential equations. Stochastic ordinary differential equations (sodes) on finite-dimensional manifolds generate stochastic flows on the manifold. One objective of the research is to construct invariant manifolds for such flows near stationary solutions, under suitable regularity and growth conditions on the driving vector fields. In particular, the investigator will prove the existence of stable, unstable and center manifolds near each stationary point on the manifold. In order for sodes to constitute viable physical models, the investigator conjectures that a Kupka-Smale type theorem must hold. An important class of infinite-dimensional semiflows on Hilbert space is generated by dissipative semilinear stochastic partial differential equations (spdes) on smooth compact manifolds. For these semiflows, the investigator intends to prove the existence of finite-dimensional invariant manifolds near stationary points. Important examples of spdes covered by this analysis are Burger's equation, affine linear stochastic evolution equations and stochastic reaction-diffusion equations. The results of the research will reveal new features of the stochastic dynamics of these well-studied models. One encounters models of stochastic systems with memory (sfdes) in many engineering and physical applications. Deterministic smooth constraints on the solutions of such models lead naturally to sfdes on (compact) Riemannian manifolds. The investigator will study the path-space-valued Markov process generated by trajectories of sfdes on the manifold. The investigator will establish an Ito formula for the infinite-dimensional segment process and will attempt to obtain its infinitesimal generator in terms of geometric invariants of the ambient Riemannian manifold. From the pathwise dynamical point of view, the investigator will construct perfect semiflows on the path space which are induced by solutions of sfdes on a compact Riemannian manifold. The existence of invariant manifolds will also be examined.The project focuses on qualitative and long-term behavior of a large class of probabilistic models known as stochastic differential equations. These equations are widely used by scientists and engineers. Of special interest is a class of models that are used in physics, engineering and biology in order to analyze dynamical systems whose evolution is influenced by random fluctuations and past history. These models are very important in a variety of diverse areas such as signal processing, stock market fluctuations, economic and labor models, aircraft dynamics, materials with memory, population dynamics and fluid flow. The investigator will use the most current probabilistic techniques in order to develop a deeper understanding of these models. The outcome of the research in this project will yield precise information on the long-term behavior of solutions of the underlying stochastic equations on states that are near statistical equilibria. Numerical algorithms will be developed whereby real market data will be used to test option-pricing models where the stock price is governed by its past history. In a different direction, the research will solidify important connections with other areas of modern mathematical research, in particular the theory of dynamical systems and geometry. The results of the project will be compiled in a research monograph targeting graduate students majoring in mathematics, engineering, and finance.