Asymptotic Problems for Stochastic Process and Differential Equations

研究者研究了动力系统中与大偏差有关的平均原理和效应。他考虑具有守恒定律的系统的扰动，特别是哈密顿系统。如果一个运动的积分有多个阱，那么慢变量的演化就会发生在一个图上，或者，在有几个第一积分的情况下，发生在一本打开的书上，一般来说，甚至纯粹的确定性扰动也会导致极限慢运动的随机性。在这种情况下，研究者研究极限随机过程，特别是，他引入了一个相对熵和研究大偏差。这允许描述不可压缩的3d流体的运动，它接近于平面运动和这种流体中的反应-扩散-对流。研究者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一个系统的相对小的确定性和随机扰动引起的长期效应。研究者考虑的问题，其中不同的过程在系统中有不同的速率（快速和缓慢的过程）。这允许对长时间行为进行一定的简化和有效的描述。研究者考虑的另一种方法涉及系统，其中长时间的行为是由其典型行为的小概率偏差来定义的。这些偏差在每个固定的时间间隔内发生的概率很小，但它们迟早会发生，并决定了系统的长时间演化。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.