Long time influence of small perturbations

这个研究项目包含了一种新的方法来解决一类重要的问题，这些问题涉及各种系统的确定性或随机扰动的长期影响，以及对这些系统的亚稳态的分析。直观地说，亚稳态行为包括系统在接近平衡状态之间的突然转变，间隔很长时间。这种行为在许多领域都表现出来，例如分子动力学、基因突变、某些经济模型等。小扰动的长期影响是微扰理论的重要组成部分，微扰理论是应用数学的主要方法之一。对于不同的时间尺度，研究和描述了扰动系统的主要状态，以及这些状态之间的转换。这些结果使得引入相对于扰动的状态稳定性度量和考虑某些最优控制问题成为可能。该项目涉及几名研究生与首席研究员一起进行相关研究课题。项目方法是基于考虑无扰动系统的不变概率测度的单纯形上的极限运动（在适当的时间尺度上）。在随机扰动的情况下，循环的层次和亚稳分布将使用大偏差型的结果来计算。这些计算将基于相应偏微分方程的非线性谱问题。如果非摄动系统有第一个积分，则其构造在某种意义上是平均原理的推广；另一方面，将在比经典情况更复杂的空间中考虑这种特定情况的有限缓慢演化。这些结果将导致对许多新效应的描述，例如纯确定性系统长期行为中的随机性。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。

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.