Degenerate diffusions in finite and infinite dimensions: smoothing and convergence

在时间上随机演化的动力学被用来描述金融市场、动荡和复杂的生物系统的行为。此外，它们在大数据算法中发挥着关键作用。在许多这样的背景下，随机性只通过空间中有限数量的方向进入系统。这种随机性的稀疏性和方程往往高维或无限维的性质导致了许多数学挑战，从获得动力学的基本结构属性到建立动力学的更精细方面。这个项目将探索有限的随机性如何与非线性相互作用以产生平滑，从而导致解和均衡的存在标准。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的性质，特别是当系统中的某些参数被移除时，系统收敛到平衡的速度。本文的目的是了解流体中的反常耗散现象。计划中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在线跨大学研究性阅读小组和爱荷华州立大学的概率阅读小组。这个项目涵盖了随机分析和动力系统交界处的几个主题。我们将研究高维和无限维简并噪声的扩散问题。一个关键的目标是了解噪声如何与非线性相互作用来产生平滑，在某种意义上，动力学属于正时间的改进的索波列夫空间，以及收敛，在某种意义上，系统在大时间内进入唯一的统计稳定状态。这种理解是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)的基础，但在湍流和统计力学中的许多重要模型中都缺乏这种理解。这类系统的更精细性质(例如，不规则点的分类、无限维亚椭圆性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)，而其他方程是简化模型，以帮助建立对激励方程的理解。这项工作将建立在获奖者之前的研究基础上，通过开发方法来分析系统的大的和有界的相空间价值。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.