Degenerate diffusions in finite and infinite dimensions: smoothing and convergence

随时间随机演化的动力学被用来描述金融市场、动荡和复杂生物系统的行为。此外，它们在大数据算法中起着关键作用。在许多情况下，随机性只能通过有限的空间方向进入系统。这种随机性的稀疏性和方程的高维或无限维性质导致了许多数学上的挑战，从获得的基本结构特性到建立动力学的精细方面。本项目将探讨有限随机性如何与非线性相互作用以产生平滑，从而导致解和平衡点存在的标准。这项工作将集中在具有随机强迫项的流体模型上，在这些模型中，人们对这种行为知之甚少。还将研究随机动力学的更精细特性，特别是当系统中的某些参数被移除时，系统收敛到平衡状态的速度有多快。这方面的目的是了解流体中的反常耗散现象。计划的工作将涉及博士后研究人员、研究生和本科生。进一步的协同活动将包括一个在线跨大学研究阅读小组和爱荷华州立大学的概率阅读小组。这个项目包含了随机分析和动力系统界面的几个主题。本文将研究高维和无限维的简并噪声扩散。关键目标是了解噪声如何与非线性相互作用以产生平滑，即动态在正时间属于改进的Sobolev空间，以及收敛，即系统在大时间内进入唯一的统计稳定状态。这种理解对于推导解的基本结构性质（例如相关拓扑中的大时间存在性/遍历性）是至关重要的，但在湍流和统计力学的许多重要模型中却缺乏这种理解。这些系统的更精细的性质（例如不规则点的分类，无限维中“半椭圆”的含义，以及反常耗散现象）也将被研究。许多要研究的方程是物理模型（例如二维随机强迫和阻尼欧拉方程或二阶朗之万动力学），而其他的是简化模型，以帮助建立对激励方程的理解。这项工作将建立在获奖者以前的研究基础上，通过开发方法来分析相空间的大值和有界值的系统。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamics that evolve randomly in time are used to describe the behavior of financial markets, turbulence, and complex biological systems. Furthermore, they play a key role in big data algorithms. In many of these contexts, randomness enters the system only through a limited number of directions in space. This sparsity of randomness and the often high or infinite-dimensional nature of the equations lead to numerous mathematical challenges, from obtaining basic structural properties of to establishing finer aspects of the dynamics. This project will explore how the limited randomness interacts with nonlinearity to produce smoothing, leading to criteria for the existence of solutions and equilibria. The work will focus on fluid models with random forcing terms, where very little is known about such behavior. Finer properties of the stochastic dynamics will also be studied, especially how fast the system converges to equilibrium as certain parameters in the system are removed. The aim in this context is to understand the anomalous dissipation phenomenon in fluids. The planned work will involve postdoctoral researchers, graduate students, and undergraduate students. Further synergistic activities will include an online cross-university research reading group and a probability reading group at Iowa State University. This project encompasses several topics at the interface of stochastic analysis and dynamical systems. Diffusions with degenerate noise in both high and infinite dimensions will be studied. A key goal is to understand how noise interacts with nonlinearities to produce smoothing, in the sense that the dynamics belongs to an improved Sobolev space for positive times, and convergence, in the sense that the system settles into a unique statistically steady state for large times. Such understanding is fundamental for deducing basic structural properties of solutions (e.g. large-time existence/ergodicity in the relevant topology), yet it is absent in many important models in turbulence and statistical mechanics. Finer properties of such systems (e.g. classification of irregular points, the meaning of "hypoellipticty" in infinite dimensions, and the anomalous dissipation phenomenon) will also be studied. Many of the equations to be investigated are physical models (e.g. the two-dimensional stochastically forced and damped Euler equations or second-order Langevin dynamics), while others are simplified models to help build understanding in the motivating equations. This work will build on previous research of the awardee by developing methods to analyze the systems for both large and bounded values of the phase space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.