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.
Isaac Sonin UNC Charlotte
Title: Insertion – A New Operation for Markov Chains and their Applications
Abstract: It is well known that a Markov chain (MC), observed only when it is outside of a subset D, is again a MC, called the Censored MC, with a new transition matrix. This matrix can be obtained in |D| iterations, each requiring O(n^2) operations, when the states from D are “eliminated” one at a time. We show how to modify these iterations to allow for a state, previously eliminated, to be “reinserted” into the state space in one iteration. This modification sheds a new light on the relationship between an initial and censored MC, and introduces a new operation – “insertion” into the theory of MCs. We briefly describe the applications of these operations in different probability models, and even outside of Probability Theory.