Stanislav Molchanov, UNC Charlotte
Title: The collisions between random and non-random sequences
Mark Freidlin, University of Maryland
Title: Long-Time Influence of Small Perturbations
Abstract: Long-time influence of small deterministic and stochastic perturbations can be described as a motion on the simplex of invariant probability measures of the non-perturbed system. I will demonstrate this general approach in the case of perturbations of a stochastic system with multiple stationary regimes. If the system has a first integral, the long–time behavior of the perturbed system, in an appropriate time scale, can be described by a motion on the Reeb graph of the first integral. This is a modified (because of the interior vertices of the Reeb graph) averaging-principle-type result. If the non-perturbed stochastic system has just a finite number of ergodic invariant probability measures, the long-time behavior is defined by limit theorems for large deviations.
Isaac Sonin, UNC Charlotte
Title: The test and find problem
Abstract: We consider the following problem: k objects (balls) are allocated at random among n boxes (sites) according to some initial distribution π, no more than one object to a box. A Decision Maker (DM) can and will test all boxes. The test is not perfect, i.e. it can give false positive and false negative results. DM has m “tags”, 1 ≤ m ≤ n, and after testing all boxes she can place l tags, 0 ≤ l ≤ m on boxes she thinks have hidden objects. She is rewarded (paid) c_i for the correct guess and penalized by d_i for a wrong guess in box i. DM knows all the testing and cost parameters of the model and her goal is to maximize the expected reward. This problem can be easily generalized into a more general problem with three or more kinds of tags: “ball is here, ball is not here, not sure, etc”, and correspondingly with a more general cost structure. This problem can be classified as a problem from Statistical Decision Theory or a problem from Search Theory or as a problem from Mathematical Statistics, or even as a discrete version of an inverse problem if parameters are not completely known to DM. The origin of this model is related to the Locks, Bombs and Testing model that we discussed at the Probability seminar last semester, but the talk will be self contained. The model is quite elementary and can be understood by every graduate and even undergraduate student interested in Applied Probability and Statistics.