Asymptotic Problems for Stochastic Process and Differential Equations

研究人员研究了动力系统中与大偏差有关的平均原理和影响。他考虑了具有守恒定律的系统的摄动，特别是哈密顿系统的摄动。如果运动积分有不止一个井，则流变量的演化发生在图上，或者，在几个第一积分的情况下，在打开的书上，甚至是纯粹的确定性摄动，通常导致极限慢运动的随机性。在这种情况下，研究人员研究了极限随机过程，特别是引入了一个相对熵，并研究了大偏差。这使得描述不可压缩三维流体在这种流体中的运动接近平面运动和反应-扩散-对流。研究人员还研究了随机共振和亚稳态作为大偏差的表现。在这个方案中，考虑了系统相对较小的确定性和随机性扰动引起的长时间效应，调查者考虑了系统中各种过程具有不同的速率(快过程和慢过程)的问题。研究人员考虑的另一种方法涉及系统，其中长时间行为由与其典型行为的小概率偏差来定义。这些偏差在每个固定时间间隔上的概率非常小，迟早会发生，并定义了系统的长期演化。这类问题出现在许多技术、物理和生物系统以及社会学模型中。

The investigator studies the averaging principle and effects relatedto large deviations in dynamical systems. He considers perturbations ofsystems with conservation laws, in particular, of Hamiltonian systems.If an integral of motion has more than one well, the evolution of theslow variables occur on a graph or, in the case of several firstintegrals, on an open book, and even pure deterministic perturbations,in general, lead to stochasticity of the limiting slow motion. In thiscase the investigator studies the limiting stochastic process, inparticular, he introduces a relative entropy and studies large deviations.This allows to describe the motion of an incompressible 3D-fluid which isclose to a planar motion and reaction-diffusion-convection in such afluid. The investigators studies also stochastic resonance andmetastability as manifestation of large deviations. Long-time effects caused by relatively small deterministic andstochastic perturbations of a system are considered in this proposal.The investigator considers problems, where various processes in the systemhave different rates (fast and slow processes). This allows certainsimplification and efficient description of the long-time behavior.Another approach considered by the investigator concerns systems, wherelong time behavior is defined by small probability deviations from theirtypical behavior. These deviations, which have very small probability oneach fixed time interval, occur sooner or later and define long timeevolution of the system. Problems of this type arise in many technical,physical, and biological systems as well as in sociological models.