Stochastic Dynamical Systems in Finite and Infinite-Dimensions

PI将研究三种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（sode's），（2）具有长记忆的约束随机微分系统和（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）解的邻域中开发基本随机流的几乎必然行为的完整表征。小扰动的随机流的几乎肯定的定性结构的影响将研究附近的双曲稳定的解决方案。这种小的扰动是自然的，因为在估计物理模型的参数时不可避免的统计误差，而在测量真实的数据时实验不准确。通用性和局部稳定性的问题将得到解决。在第二类问题中，将识别出一类正则的具有长记忆的约束随机系统。这样的类允许存在光滑的随机半流，并因此使用适当修改的遍历理论技术的不变流形的特征。几何约束和随机动力学之间的相互作用将被检查。弱和强逼近计划将开发的随机系统与全记忆，然后应用到期权定价模型在数学金融与延迟股票动态。第三类问题的动力学将通过分析经典的例子，如二维随机Navier-Stokes方程和Burgers方程进行研究。拟议的研究是一项长期计划，倡导概率论/随机分析与传统主流数学学科（如动力系统，微分几何和数值分析）之间的新联系。在过去的十年中，相当多的应用数学家、工程师和经济学家把他们的注意力转向了随机演化记忆系统，用于模拟各种物理现象，这些物理现象的时间演化依赖于它们过去的历史。在物理学中，经常研究具有延迟反馈的激光动力学，以及具有延迟的噪声双稳态系统的动力学。在生物物理学中，具有记忆的随机（即随机）系统被用于模拟延迟视觉反馈系统或人体姿势摇摆，以及设计心脏起搏器细胞。在数学金融学中，股票的波动性可能取决于其过去的历史，因此股票动态可能最好由具有记忆的随机系统来描述。该项目的研究有望对一类称为随机偏微分方程的无穷维模型的平衡点附近的稳定性结构给出完整的表征。这种模型在热流、流体运动和气候变化模型的研究中普遍存在。研究所将完成关于具有记忆的随机系统的研究专著的编写工作。该专着旨在成为基础的研究生课程在数学卡本代尔。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.