Asymptotic Problems for Stochastic Process and Differential Equations

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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.