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

随机性在现代科学技术中无处不在。它是空气湍流模型、化学动力学和大数据算法的基础。这种模型通常是高维的，甚至是无限维的，就像湍流的情况一样。理解这些系统的核心是理解能量和随机性通过动力学中的非线性相互作用传播的方式。这个研究项目探索了许多方向，利用这些系统的几何和代数结构来更好地理解这些基本的传输现象。研究人员将努力了解这种相互作用是如何在缺乏随机性的不稳定系统中导致稳定的。该项目包括研究生、本科生和高中生的研究参与。在高中阶段，调查人员还将努力让学生的老师与前沿数学保持联系，帮助他们成为更有效、更灵通的教师。此外，我们将与纪录片中心合作，制作录影记录学生的研究工作，以进一步传播经验和扩大计划的影响。这个在随机分析和动力学方面的合作研究项目将探索随机系统中噪声和耗散的传播及其对稳态存在和结构的影响。特别强调发生在随机偏微分方程中的效应，噪声稳定不稳定系统的能力，以及强迫系统中的非平衡稳态。许多要研究的方程是物理模型（例如来自流体力学或非平衡统计物理），而其他要研究的方程作为示例，有助于理解这些系统中产生稳定或不稳定的潜在机制。这项工作将建立在研究者之前在有限维随机常微分方程（SODEs）中设计Lyapunov函数的工作基础上，并建立实用的方法来证明在有限维随机常微分方程（SODEs）和无限维随机常微分方程（SPDEs）中唯一的遍历性和收敛性。虽然这项研究预计将对流体力学和统计物理学产生直接影响，但研究这种系统所产生的技术将适用于其他领域的广泛问题，如生物学、工程学、物理学和金融学，这些领域通常采用随机常微分方程和偏微分方程作为建模工具。

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.