Abstract:
We describe metastability for randomly perturbed degenerate diffusion processes in a way that is parallel to the Freidlin-Wentzell theory for randomly perturbed dynamical systems with finitely many invariant manifolds. In a more abstract setting, metastability can be viewed as an extension of the ergodic theorem for Markov chains to the case of parameter-dependent semi-Markov processes. Stated in PDE terms, the problems concern the asymptotic behavior of solutions to parabolic equations whose coefficients degenerate at the boundary of a domain. The operator may be regularized by adding a small diffusion term. Metastability effects arise in this case: the asymptotics of solutions, as the size of the perturbation tends to zero, depends on the time scale. Initial-boundary value problems with both the Dirichlet and the Neumann boundary conditions are considered. We also consider periodic homogenization for operators with degeneration. The talk is based on joint work with M. Freidlin.
Biography:
Leonid Koralov is a professor in the Department of Mathematics at the University of Maryland, where he has been a faculty member since 2005. He served as an assistant professor from 2005 to 2008, then as an associate professor until becoming a full professor in 2012. Prior to joining the University of Maryland, Professor Koralov was an assistant professor at Princeton University and a member of the Institute for Advanced Study in Princeton. He obtained his Ph.D. from SUNY at Stony Brook in 1998 and holds a BSc from Moscow State University. Koralov works in the areas of stochastic processes, partial differential equations, and dynamical systems, with particular interest in the asymptotic analysis of probabilistic models, including metastability and homogenization.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
This event is open to the NYU Shanghai community and Math community.