Martin Day – Description of Research Interests

My recent interests concern the structure of differential games and control problems arising in the fluid or risk-sensitive / large deviations limit of queueing networks.  This produces interesting deterministic optimization problems, which can be solved explicitly in a number of (simple) examples.

Much of my work in the past concerns the deterministic optimization problems associated with large-deviations limits of small random perturbations of stable dynamical systems.  Many physical systems (e.g. mechanical/electronic devices, biological or communications systems) can be described by what mathematicians call a "dynamical system". Such a mathematical description is usually only an idealization. It typically oversimplifies some aspects of the real situation, and hence fails to predict some features of its behavior. In particular the mathematical description may not account for the small unpredictable (i.e. random) disturbances that are present in any real situation, such as noise in a radio transmission, or small variations from the usual pattern of incoming calls to the telephone system. Much of my past research has been concerned with mathematical issues involved in studying how the behavior of a dynamical system is altered by the presence of such small random influences, especially those situations in which the random influences cause the occurrence of some phenomenon that would never occur in the idealized dynamical system itself.

The effect of introducing random influences into a dynamical system produces what is called a stochastic process. (Examples are Markov chains, queueing processes or solutions of stochastic differential equations.) The study of such stochastic processes is a branch of probability theory that has some very close connections to parts of analysis, including partial and ordinary differential equations. The problems of small random perturbations in particular have important connections to the calculus of variations, classical mechanics and control on nonlinear systems. My work is motivated by the insight these connections can bring to the mathematical problems, as well as the implications for various situations in the applied sciences. 

My publications are listed on a separate page, where you will find links to preprint draft versions of more recent papers.