Second-Order Investigations in Particle and Trapping Systems

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The goal of this research is to investigate the fluctuation behaviors in two types of systems: Mass conservative particle systems, and Brownian trapping models. The first type includes the exclusion and zero-range processes, while the second type covers variants of the "Wiener Sausage" model. Three categories of problems are focused upon in these systems: Tagged particle problems in conservative particle systems, current fluctuations at a fixed location in the exclusion model, and localization structure of long time surviving Brownian paths in trap settings.Stochastic dynamical systems following the motion of a collection of particles have been successful in the modeling of diverse physical phenomena such as fluid and traffic flow, queuing, chemical trapping behavior, etc. Several theoretical studies have been made on "first-order" calculations, which describe the dominant modes of objects of interest, for instance, average velocities or trap survival probabilities. The intent of this project is to supply the "second-order" characteristics in these fluid and traffic models. The significance of second-order calculations is that they provide a measure of the robustness of the first-order approximations in applications. Aside from this general notion, the computations involved in this project would shed light on the intrinsic physics of the random media models studied. Also, from the mathematical view, the solution techniques of these challenging problems would be of great value themselves for future endeavors.