Long time influence of small perturbations

小扰动的长期影响

• 批准号: 2307377
• 负责人: Leonid Koralov
• 金额: $ 20万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307377&HistoricalAwards=false
• 关键词: Long; time; influence; small; perturbations

This research project contains a new approach to an important class of problems concerning the long-term influence of deterministic or random perturbations of various systems and to the analysis of metastability of such systems. Intuitively, metastable behavior includes sudden transitions, separated by long time intervals, between near-equilibrium states of a system. This behavior is exhibited inside a variety of fields, e.g., in molecular dynamics, genetic mutations, certain economic models, etc. The long-term influence of small perturbations is an essential part of the perturbation theory - one of the main methods in applied mathematics. For various time scales, the main states of the perturbed system are investigated and described, along with the transitions between these states. These results enable introducing a measure of stability of the states with respect to the perturbations and considering certain optimal control problems. The project involves several graduate students working along with the principal investigator on related research topics. The project approach is based on considering the limiting (in an appropriate time scale) motion on the simplex of invariant probability measures of the unperturbed system. In the case of random perturbations, the hierarchy of cycles and metastable distributions will be calculated using large-deviation-type results. These calculations will be based on a non-linear spectral problem for the corresponding partial differential equations. If the unperturbed system has a first integral, the construction is, in a sense, a generalization of the averaging principle; on the other hand, the limiting slow evolution for this specific case will be considered on a space that is more sophisticated than in the classical case. The results will lead to a description of a number of new effects, such as stochasticity in the long-time behavior of purely deterministic systems.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.

该研究项目包含一种针对重要问题的新方法，即确定性或随机扰动的长期影响，以及对此类系统的亚稳定性的分析。直观地，亚稳态的行为包括突然的过渡，在系统的近平衡状态之间被长时间间隔隔开。这种行为在各个领域内表现出，例如，在分子动力学，遗传突变，某些经济模型等中。小扰动的长期影响是扰动理论的重要组成部分，这是应用数学的主要方法之一。对于各种时间尺度，研究并描述了这些状态之间的过渡，研究并描述了扰动系统的主要状态。这些结果使得能够在扰动方面引入状态的稳定性度量，并考虑某些最佳控制问题。该项目涉及几名研究生与有关相关研究主题的主要研究者一起工作。 该项目方法基于考虑不受干扰系统不变概率度量的限制（以适当的时间尺度）运动。在随机扰动的情况下，将使用大型viriation-type结果计算循环和亚稳态分布的层次结构。这些计算将基于相应的部分微分方程的非线性光谱问题。如果不受干扰的系统具有第一个组成部分，则在某种意义上，结构是平均原理的概括。另一方面，这种特定情况的限制缓慢演变将在比经典情况更复杂的空间上考虑。结果将导致对许多新效果的描述，例如纯粹确定性系统的长期行为中的随机性。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来支持的。