Leonid Koralov, University of Maryland College Park

*Title: *Large Time Behavior of Randomly Perturbed Dynamical Systems

*Abstract: *We will discuss several asymptotic problems for randomly perturbed flows (and related problems for Markov chains with rare transitions). One class of flows (with regions where a strong flow creates a trapping mechanism) leads to a new class of elliptic and parabolic boundary value problems with non-standard boundary conditions. The same boundary value problems appear as a limiting object when studying the asymptotic behavior of diffusion processes with pockets of large diffusivity.

We will also discuss how large-deviation techniques can be used to study the asymptotic behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of the corresponding diffusion processes.