|Professor: Leonid Koralov, Department of Mathematics, University of Maryland|
|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.