Asymptotic Problems for Stochastic Process and Differential Equations

研究了动力系统中的平均原理和大偏差效应.他认为扰动系统的守恒定律，特别是汉密尔顿系统。如果一个积分的运动有一个以上的井，演变的theslow变量发生在一个图形，或在案件的几个第一积分，在一本打开的书，甚至纯粹的确定性扰动，一般来说，导致随机性的限制慢动作。在这种情况下，研究者研究了极限随机过程，特别是，他引入了相对熵并研究了大偏差，这允许描述不可压缩三维流体的运动，它接近于平面运动和反应-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。 在这个建议中考虑了由系统的相对较小的确定性和随机扰动引起的长时间效应，研究者考虑了系统中各种过程具有不同速率（快过程和慢过程）的问题。这使得对长时间行为的描述更加简单和有效。研究者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的时间间隔上都有很小的概率，它们迟早会发生，并定义了系统的长期演化。这种类型的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.