Long-term Effects of Small Perturbations and Other Multiscale Asymptotic Problems

微小扰动的长期影响，无论是确定性的还是随机性的，在现实生活中和从历史、社会学、经济学到生物学、物理学和工程学的各种智力企业都非常感兴趣。当一个人想要了解一个复杂系统的时间演化时，一个可以分析的系统的简化模型通常是一个前进的方向。在这种情况下，人们选择了相对较少的引导系统演变的主要‘因素’或变量，而忽略了其他相对不重要的因素。在很短的时间间隔内，这些小因素并不是必不可少的。然而，在很长的时间尺度上，那些被认为可以忽略不计的因素对于决定系统的行为可能变得重要，甚至是关键的。这位研究人员和他的同事们正在开发研究这些问题的新模型和方法。“问题”的例子包括气候变化、生物进化和物理或经济系统中的相变，以及由微小扰动引起的“稳定”振荡、平衡和其他模式的出现。研究人员和他的同事研究了各种动力系统的确定性和随机扰动，以及与偏微分方程演化相对应的半流。乍一看，对不同的问题提出了一种通用的方法。作为大偏差理论的表现形式的亚稳性和随机共振，以及平均化原理及其修正，都可以从这个角度来考虑。这一一般方法是基于将极限慢演化作为在非摄动系统的不变测度的锥体上的运动来研究的。该方法强调了联合考虑确定性和随机性扰动的重要性。用这种方法可以研究分子马达模型中产生的多自旋广义Landau-Lifshitz方程、三维中具有守恒律的不可压缩流动、窄通道中的反应-平流-扩散方程。具有小参数的二阶椭圆型方程的各种边值问题也可以考虑。最后，PI还将研究朗之万方程的小质量渐近性。

Long term effects of small perturbations, both deterministic and stochastic are of great interest in real life and for various intellectual enterprises from history, sociology, and economics to biology, physics, and engineering. When one wants to understand the time evolution of a complex system, a simplified model of the system, that can be analyzed, is usually a way forward. In this case one chooses a relatively small number of main 'factors' or variables which are guiding the evolution of the system while neglecting other factors that are relatively insignificant. On a short time interval, these small factors are not essential. However, on long time scales, the factors, which were considered as negligible, can become important and even critical for determining the system's behavior. The investigator and his colleagues are developing new models and methods for studying such problems. 'Problem' examples include climate change, biological evolution, and phase transitions in physical or economic systems as well as the appearance of 'stable' oscillations, equilibriums, and other patterns caused by small perturbations.The investigator and his colleagues study deterministic and stochastic perturbations of various dynamical systems as well as semi-flows corresponding to the evolution of PDEs. A general approach to, at first glance, different problems is suggested. Metastability and stochastic resonance, which are manifestation of the large deviation theory, as well as the averaging principle and its modifications can be considered from this point of view. This general approach is based on the study of limiting slow evolution as a motion on the cone of invariant measures of the non-perturbed system. The approach underlines the importance of joint consideration of deterministic and stochastic perturbations. Perturbations of the generalized Landau-Lifshitz equation for multiple spins, incompressible flows in 3D having a conservation law, reaction-advection-diffusion equations in narrow channels arising in models of molecular motors can be studied using this approach. Various boundary problems for second order elliptic equations with a small parameter can be considered as well. Finally, the PI will also study the small mass asymptotics for the Langevin equation.