Stochastic Dynamical Systems in Finite and Infinite-Dimensions

PI将研究三类不同类型的微分系统的随机动力学：(1)光滑约束下的随机常微分方程组(SODE‘s)，(2)长记忆约束随机微分方程组(2)和(3)随机偏微分方程(SPDE’s)。在第一类问题中，PI将在平稳(非遍历)解的邻域内发展基本随机流的几乎必然行为的完整特征。我们将在双曲定常解附近研究小扰动对随机流几乎必然定性结构的影响。这种微小的扰动是自然的，因为在估计物理模型的参数时不可避免地存在统计误差，而在测量真实数据时存在实验上的不准确。将解决一般性和地方稳定性问题。在第二类问题中，将识别一类具有长记忆的正则约束随机系统。这样的类允许光滑随机半流的存在，从而使用适当修改的遍历理论技巧来刻画其不变流形。我们将研究约束几何和随机动力学之间的相互作用。建立了具有满记忆的随机系统的弱逼近和强逼近格式，并将其应用于具有时滞股票动态的数学金融中的期权定价模型。第三类问题的动力学将通过分析二维随机Navier-Stokes方程和Burgers方程等经典例子来研究。这项拟议的研究是一项长期计划，倡导概率论/随机分析与动力系统、微分几何和数值分析等传统主流数学学科之间的新联系。在过去的十年里，相当多的应用数学家、工程师和经济学家将他们的注意力转向随机演化的具有记忆的系统，以模拟各种物理现象，这些物理现象的时间演化取决于它们过去的历史。在物理学中，人们经常研究具有延迟反馈的激光动力学，以及含噪声的时滞双稳系统的动力学。在生物物理学中，随机(即具有记忆的随机)系统用于对延迟视觉反馈系统或人体姿势摆动进行建模，并用于心脏起搏器细胞的设计。在数学金融学中，股票的波动性可能依赖于它过去的历史，因此股票的动态可能最好地用一个有记忆的随机系统来描述。这个项目的研究有望给出一大类称为随机偏微分方程的无限维模型平衡点附近的稳定性结构的完整刻画。这种模型在研究热流、流体运动和气候变化的模拟中普遍存在。PI将完成一本具有记忆的随机系统研究专著的准备工作。这本专著旨在成为卡本代尔大学数学研究生课程的基础。具有长记忆的随机系统及其在数学金融期权定价中的应用将吸引PI的硕士和博士研究生，其中一些是女性和少数民族。

The PI will study the stochastic dynamics of three different classes of differential systems: (1) stochastic ordinary differential equations (sode's) under smooth constraints, (2) constrained stochastic differential systems with long memory and (3) stochastic partial differential equations (spde's). In the first class of problems, the PI will develop a complete characterization of the almost sure behavior of the underlying stochastic flow in the neighborhood of a stationary (non-ergodic) solution. The effect of small perturbations on the almost sure qualitative structure of the stochastic flow will be studied near hyperbolic stationary solutions. Such small perturbations are natural because of unavoidable statistical errors in estimating the parameters of physical models against experimental inaccuracies in the measurement of real data. Issues of genericity and local stability will be addressed. In the second class of problems, a regular class of constrained stochastic systems with long memory will be identified. Such classes allow for the existence of smooth stochastic semiflows and hence a characterization of their invariant manifolds using suitably-modified ergodic theory techniques. The interplay between the geometry of the constraints and the stochastic dynamics will be examined. Weak and strong approximation schemes will be developed for stochastic systems with full memory and then applied to option-pricing models in mathematical finance with delayed stock-dynamics. The dynamics of the third class of problems will be studied by analyzing classical examples such as two-dimensional stochastic Navier-Stokes and Burgers equations. The proposed research is a long-term program that advocates novel links between probability theory/stochastic analysis and traditional mainstream mathematical disciplines such as dynamical systems, differential geometry and numerical analysis. In particular, the research would lead to new interactions between stochastic geometry and dynamical systems.During the past decade, a considerable number of applied mathematicians, engineers and economists have turned their attention to randomly evolving systems with memory for modeling a variety of physical phenomena whose time evolution depends on their past history. In physics, laser dynamics with delayed feedback is often investigated, as well as the dynamics of noisy bi-stable systems with delay. In biophysics, random (viz. stochastic) systems with memory are used to model delayed visual feedback systems or human postural sway and in the design of cardiac pacemaker cells. In mathematical finance, the volatility of the stock may be dependent on its past history and hence the stock dynamics may be best described by a stochastic system with memory. The research in this project is expected to give a complete characterization of the stability structure near equilibria for a large class of infinite-dimensional models called stochastic partial differential equations. Such models are ubiquitous in the study of heat flow, the movement of fluids and modelling of climate change.The PI will complete the preparation of a research monograph on stochastic systems with memory. The monograph is intended to be the basis for a graduate course in mathematics at Carbondale. Stochastic systems with long-memory and their applications to option-pricing in mathematical finance will engage the PI's master's and doctoral graduate students, some of them are females and minorities.