Asymptotic Problems for Stochastic Processes and PDE's

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Several classes of asymptotic problems are considered. The averaging principle for deterministic and stochastic perturbations, asymptotic problems for reaction-diffusion equations, and problems related to stochastic resonance are among them. The long-time evolution of perturbed systems with conservation laws, even in the case of purely deterministic perturbations, leads, in general, to stochastic process on complexes defined by the conservation laws. So the classical averaging principle (say, for deterministic perturbations of integrable Hamiltonian systems when the Hamiltonian has many critical points) should be treated in the stochastic framework. In this research small diffusion asymptotics for reaction-diffusion in an incompressible 2D-fluid, which is closely related to the averaging for Hamiltonian systems, is studied. Another class of problems concerns the large deviation theory and stochastic resonance. A number of new effects such as large amplitude oscillations and stabilization induced by the small noise in autonomous systems are considered.The asymptotic approach is one of the most powerful tools of applied mathematics. In particular, the averaging principle plays the leading role when systems combining multi-scale processes are considered. Such problems arise in mechanics, in material sciences, in biophysics, and in other areas. This research does not just consider problems concerning the mathematical justification of the averaging principle, but also describes new applications and new effects. In recent years stochastic-resonance-type effects, which first appeared in the theory of long-time evolution of the climate, have attracted the attention of specialists in many areas of physics, engineering, and biology. The mathematical theory of these effects is