Asymptotic Problems for Stochastic Processes and Differential Equations

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Several classes of asymptotic problems will be considered. The first class concerns metastability and stochastic resonance for multiscale systems. The general approach to metastability and stochastic resonance in the framework of large deviations was developed in our previous papers. In the case of multiscale systems these notions should be modified. In particular, new effects appear in systems which are close to Hamiltonian and in systems with a small delay. Another class consists of new problems for PDE's with a small parameter and related stochastic processes. To describe the limiting behavior in these classical problems one should consider equations on graphs or open book spaces related to the original problam. Reaction-Diffusion-Advection equations in incompressible fluids, in strongly unisotropic media, and in narrow tubes will be considered. The last class of problems concerns the averaging principle for systems with many degrees of freedom and multiwell Hamiltonians. It turns out that even in pure deteministic systems the slow motion can be, in a sense, stochastic. Small perturbations of a system which often have negligible effects, when the system considered on a short time interval, can become crucial on larger timescale. The study of such effects is one of our main goals. We consider long time influence of small stochastic and deterministic perturbations and, under certain conditions, describe the long-time evolution of the system which essentially depends on the perturbations. Such kind of problems arise in numerous applications from physics and biology to economics and sociology. We are developing new mathematical approches to these problems.