Asymptotic Problems for Stochastic Process and Differential Equations

University Of Maryland, College Park, College Park MD

Investigators

Abstract

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