Collaborative Research: Propagation of Dissipation: Stochastic Stabilization in Finite and Infinite Dimensions

合作研究：耗散传播：有限和无限维中的随机稳定

基本信息

批准号：
1613337
负责人：
Jonathan Mattingly
金额：
$ 22.86万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-06-15 至 2021-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613337&HistoricalAwards=false
关键词：
Collaborative Research Propagation Dissipation Stochastic

项目摘要

Randomness is everywhere in modern science and technology. It underlies models of air turbulence, chemical dynamics, and algorithms for big data. Such models are often high dimensional or even infinite dimensional, as in the case of turbulence. Central to understanding these systems is understanding the way both energy and randomness are spread by nonlinear interactions in the dynamics. This research project explores a number of directions that exploit both the geometric and algebraic structures of such systems to better understand these basic transport phenomena. The investigators will work to understand how such interactions can lead to stabilization in systems that are unstable in the absence of randomness. The project includes research participation by graduate, undergraduate, and high-school students. At the high school level, the investigators will also work to keep the students' teachers connected with cutting-edge mathematics, helping them to be more effective and better informed as teachers. In addition, videos chronicling the student research efforts will be made in collaboration with a center for documentary films to further disseminate the experience and broaden the project's impact. This collaborative research project in stochastic analysis and dynamics will explore the propagation of noise and dissipation in stochastic systems and their effects on the existence and structure of stationary states. There is particular emphasis on effects that occur in stochastic partial differential equations, the ability of noise to stabilize unstable systems, and non-equilibrium steady-states in forced systems. Many of the equations to be investigated are physical models (e.g. from fluid mechanics or non-equilibrium statistical physics) while other equations to be studied serve as examples that aid in understanding the underlying mechanisms producing stability or instability in such systems. This work will build on the investigators' previous work in designing Lyapunov functions in the finite-dimensional setting of stochastic ordinary differential equations (SODEs) and establishing practical methods for proving unique ergodicity and convergence to equilibrium in both the finite-dimensional setting of SODEs as well as the infinite-dimensional setting of SPDEs. While the research is anticipated to have immediate implications in fluid mechanics and statistical physics, the techniques that will result from studying such systems will be applicable to a wide family of problems in other areas, such as biology, engineering, physics, and finance, which regularly employ stochastic ordinary and partial differential equations as modeling tools.

在现代科学和技术中，随机性无处不在。它是大数据的空气湍流，化学动力学和算法模型的基础。这种模型通常是高维甚至无限的尺寸，例如湍流。理解这些系统的核心是了解动态中非线性相互作用传播能量和随机性的方式。该研究项目探讨了许多方向，这些方向利用了此类系统的几何和代数结构，以更好地理解这些基本运输现象。研究人员将努力了解这种相互作用如何导致在没有随机性的情况下不稳定的系统中稳定。该项目包括研究生，本科和高中生的研究参与。在高中，调查人员还将努力使学生的老师与先进的数学建立联系，从而帮助他们更加有效，更了解教师。此外，还将与纪录片中心合作制作记载学生研究工作的视频，以进一步传播体验并扩大项目的影响。这项在随机分析和动态方面的协作研究项目将探讨随机系统中噪声和耗散的传播及其对固定状态的存在和结构的影响。特别强调了随机部分微分方程，噪声稳定不稳定系统的能力以及强制系统中的非平衡稳态。要研究的许多方程是物理模型（例如，来自流体力学或非平衡统计物理学），而要研究的其他方程式作为示例，有助于理解在此类系统中产生稳定性或不稳定的潜在机制。这项工作将基于调查人员以前在设计Lyapunov功能的工作中，以随机的普通微分方程（SODE）的有限维度设置，并建立实用方法，以证明在SODES的有限尺寸设置中证明独特的麦迪和融合，以使SODES和Infinite dimensiential dimensiential spdes sepses and spdess and spdess sepdes sequilibrium均衡。虽然预计该研究对流体力学和统计物理学有直接的影响，但研究此类系统将导致的技术将适用于其他领域的广泛问题，例如生物学，工程，物理和财务，这些技术经常使用随机的普通和部分差分方程作为建模工具。