Long time influence of small perturbations

该研究项目包含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此类系统的亚稳定性分析。直觉上，亚稳态行为包括系统的近平衡状态之间的突然转变，由长时间间隔分隔。这种行为表现在各种领域，例如，小扰动的长期影响是扰动理论的重要组成部分--应用数学的主要方法之一。对于不同的时间尺度，受扰系统的主要状态的调查和描述，沿着这些状态之间的过渡。这些结果使引进一个措施的稳定性的状态相对于扰动，并考虑某些最优控制问题。该项目涉及几名研究生与主要研究者一起沿着研究相关的研究课题。 该项目的方法是基于考虑限制（在适当的时间尺度）运动的单纯形不变的概率措施的未扰动系统。在随机扰动的情况下，将使用大偏差类型的结果来计算循环和亚稳分布的层次。这些计算将基于相应的偏微分方程的非线性谱问题。如果未扰动系统有一个第一积分，那么这个构造在某种意义上是平均原理的推广;另一方面，在这个特殊情况下的极限慢演化将在一个比经典情况更复杂的空间上考虑。该奖项反映了NSF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.