Random Perturbations of Excited Deterministic Systems

受激确定性系统的随机扰动

• 批准号: 1855504
• 负责人: David Herzog
• 金额: $ 17.54万
• 依托单位: Iowa State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855504&HistoricalAwards=false
• 关键词: Random; Perturbations; Excited; Deterministic; Systems

For over a century, randomness has been a core modeling tool used to describe phenomena such as, but not limited to, the value of stock prices over time, the motion of particles subject to thermal fluctuations (e.g. pollen grains or lipids suspended in water) and the time evolution of competing populations in biology. Additionally, randomness is a key component in many algorithms used to process and understand big data. Crucial to analyzing such models and algorithms is understanding precisely how the randomness and nonlinear effects, i.e. the excitation, combine to relax the system to equilibrium. The research supported by this award will seek to quantify rates of convergence to equilibrium in a number of models in statistical physics, turbulence and molecular dynamics, especially in those systems used in random sampling algorithms. The project includes collaborations with high school, undergraduate and graduate students, the results of which will be disseminated through publications and through presentations at domestic and international conferences. This award supports several projects at the interface of stochastic analysis and dynamics. The first part of this research focuses on extracting explicit rates of convergence to equilibrium for stochastic differential equations (SDEs) used in random sampling algorithms. Of particular importance is understanding the precise nature of convergence in high dimensions in the presence of singular coefficients in the equations. Such singularities arise naturally in the context of molecular dynamics (e.g. Lennard-Jones, Coloumb potential functions) as they are used to describe repulsive effects as particles approach one another. Although natural from a modeling and statistical standpoint, the singularities lead to challenges in both analyzing the dynamics itself as well as in analyzing the corresponding numerical simulation due to energy spikes generated by the singular interactions. The second part focuses on studying large-time properties of various statistical models in turbulence. In each of these models, there is also a source of energy often leading to interesting effects in equilibrium such as, for example, heavy-tailed invariant measures. Understanding how randomness, nonlinearities, dissipation as well as the excitation balance out to produce these interesting effects is of central importance.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.

一个多世纪以来，随机性一直是一种核心建模工具，用于描述诸如但不限于股票价格随时间变化的现象、受热波动影响的颗粒的运动(例如，悬浮在水中的花粉或脂类)以及生物学中相互竞争的种群的时间演变。此外，在许多用于处理和理解大数据的算法中，随机性是一个关键因素。分析这样的模型和算法的关键是准确地理解随机性和非线性效应，即激励是如何结合在一起，使系统放松到平衡的。该奖项支持的研究将寻求量化统计物理、湍流和分子动力学中的一些模型的收敛到平衡的速度，特别是在随机抽样算法中使用的那些系统中。该项目包括与高中生、本科生和研究生的合作，其结果将通过出版物和在国内和国际会议上的发言来传播。该奖项支持几个随机分析和动力学相结合的项目。本研究的第一部分集中于提取随机抽样算法中使用的随机微分方程(SDE)的显式收敛到均衡的速度。尤其重要的是，当方程中存在奇异系数时，理解高维收敛的精确性质。这种奇点在分子动力学(如Lennard-Jones，Coloumb势函数)的背景下自然产生，因为它们被用来描述粒子相互接近时的排斥效应。虽然从建模和统计的观点来看，奇异性是自然的，但由于奇异相互作用产生的能量尖峰，奇异性导致了分析动力学本身以及分析相应的数值模拟的挑战。第二部分重点研究了各种统计模型在湍流中的大时间特性。在这些模型中的每一个中，还有一种能量源经常导致在平衡中产生有趣的效应，例如，重尾不变度量。了解随机性、非线性、耗散以及激发平衡如何产生这些有趣的影响是至关重要的。该奖项反映了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。

项目成果

Sensitivity of steady states in a degenerately damped stochastic Lorenz system

简并阻尼随机洛伦兹系统稳态的灵敏度

• DOI: 10.1142/s0219493721500556
• 发表时间: 2021
• 期刊: Stochastics and Dynamics
• 影响因子: 1.1
• 作者: Földes, Juraj;Glatt-Holtz, Nathan E.;Herzog, David P.
• 通讯作者: Herzog, David P.

Gibbsian dynamics and the generalized Langevin equation

吉布斯动力学和广义朗之万方程

• DOI: 10.1214/23-ejp904
• 发表时间: 2023
• 期刊: Electronic Journal of Probability
• 影响因子: 1.4
• 作者: Herzog, David P.;Mattingly, Jonathan C.;Nguyen, Hung D.
• 通讯作者: Nguyen, Hung D.

The method of stochastic characteristics for linear second-order hypoelliptic equations

线性二阶亚椭圆方程的随机特性方法

• DOI: 10.1214/22-ps11
• 发表时间: 2023
• 期刊: Probability Surveys
• 影响因子: 1.6
• 作者: Földes, Juraj;Herzog, David P.
• 通讯作者: Herzog, David P.

Gamma Calculus Beyond Villani and Explicit Convergence Estimates for Langevin Dynamics with Singular Potentials

• DOI: 10.1007/s00205-021-01664-1
• 发表时间: 2021-06-08
• 期刊: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
• 影响因子: 2.5
• 作者: Baudoin, Fabrice;Gordina, Maria;Herzog, David P.
• 通讯作者: Herzog, David P.

A functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero

零时弱亚椭圆扩散的迭代对数泛函定律

• DOI: 10.1016/j.spa.2022.03.012
• 发表时间: 2022
• 期刊: Stochastic Processes and their Applications
• 影响因子: 1.4
• 作者: Carfagnini, Marco;Földes, Juraj;Herzog, David P.
• 通讯作者: Herzog, David P.