The principal investigator's main research on stochastic processes continues to be in the area of queueing networks, and on topics related to branching random walks and interacting particle systems. These areas deal with large systems of objects, such as customers, jobs, or particles, that are connected by some random interaction rule. He has recently also become interested in coupling problems for Brownian motion on certain bounded domains. In the area of queueing networks, an important question over the past two decades had been when such a system is stable and, when it is stable, to analyze its behavior; presently, a general theory is still lacking. The PI intends to study the behavior of several families of networks, such as those where the customer always joins the shortest queue and those connected with Internet traffic. In the area of interacting particle systems, one typically studies the evolution of large lattice-valued random systems, which model various complex random systems and exhibit a wide range of spatial behavior; they frequently occur in the context of mathematical physics and mathematical biology. These models sometimes overlap with models involving branching random walks or with other models from mathematical physics. The PI intends to study the behavior of a number of such models, including those for branching random walks, Gaussian free fields, and the exclusion process. The PI also intends to study certain properties of Brownian motion, in particular, under what conditions pairs of Brownian motions on certain bounded domains can, with positive probability, remain separated for all time.
The principal investigator's research is concentrated in several areas of probability theory. He works in the area of queueing networks, where the behavior of how lines evolve is studied. Examples include human lines ("customers") or assembly lines for component parts ("jobs") of a manufacturing process, for instance, the manufacture of semiconductor wafers. When the rule by which customers or jobs are served involves multiple steps, the evolution in time of such lines can have unexpected behavior; understanding such behavior is important for designing efficient and cost-effective rules for these systems. Much of PI's work in queueing networks is in this direction, with continuing work including systems where customers or jobs choose to join the shortest line, for instance, when shoppers pay for purchases at a supermarket. The principal investigator also performs research in the area of interacting particle systems, where the evolution of complicated random systems is studied. Such systems arise in physics and biology, and their components can correspond to particles, cells, or individual organisms, and may model, for instance, the spread of tumor cells. Because of the very different behavior exhibited by different systems, many aspects are currently only partially understood, and since such systems are typically too complicated to do explicit computations, a mathematical theory to understand these systems is important. This project will study the behavior of a number of such models.
