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，随机性一直是用于描述现象的核心建模工具，例如但不限于股票价格随时间的变化、受热波动影响的粒子运动（例如花粉粒或悬浮在水中的脂质）以及生物学中竞争种群的时间演化。 此外，随机性是许多用于处理和理解大数据的算法的关键组成部分。 分析此类模型和算法的关键是准确理解随机性和非线性效应（即激励）如何结合起来使系统松弛至平衡。 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许多模型的平衡收敛速度，特别是在随机采样算法中使用的系统中。 该项目包括与高中生、本科生和研究生的合作，其成果将通过出版物以及在国内和国际会议上的演讲来传播。 该奖项支持随机分析和动力学领域的多个项目。 本研究的第一部分重点是提取随机采样算法中使用的随机微分方程 (SDE) 收敛到平衡的显式速率。 特别重要的是理解方程中存在奇异系数的情况下高维收敛的精确性质。 这种奇点在分子动力学（例如 Lennard-Jones、哥伦布势函数）的背景下自然出现，因为它们被用来描述粒子彼此接近时的排斥效应。 尽管从建模和统计的角度来看，奇点是自然的，但由于奇点相互作用产生的能量尖峰，奇点给分析动力学本身以及分析相应的数值模拟带来了挑战。 第二部分重点研究湍流中各种统计模型的大时间特性。 在每个模型中，还有一种能量来源，通常会导致平衡中有趣的效应，例如重尾不变测度。 了解随机性、非线性、耗散以及激励平衡如何产生这些有趣的效果至关重要。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值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

Weighted L 2 -contractivity of Langevin dynamics with singular potentials

具有奇异势的 Langevin 动力学的加权 L 2 -收缩性

• DOI: 10.1088/1361-6544/ac4152
• 发表时间: 2021-12
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Camrud, Evan;Herzog, David P;Stoltz, Gabriel;Gordina, Maria
• 通讯作者: Gordina, Maria

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: 2022-07
• 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.

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: 2023-01
• 期刊: Probability Surveys
• 影响因子: 1.6
• 作者: Földes, Juraj;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: 2023-01
• 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.

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: 2021-12
• 期刊: Stochastics and Dynamics
• 影响因子: 1.1
• 作者: Földes, Juraj;Glatt;Herzog, David P.
• 通讯作者: Herzog, David P.