Mathematical Sciences: Complex Stochastic Systems

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9504323: Kurtz/Ney Abstract. Research programs will be carried out in the areas of Markov processes, stochastic analysis, chains with infinite memory, large deviation theory, branching processes, and stochastic control. Work on Markov processes will focus on measure-valued processes related to population genetics and other population models. Work on large deviations will include both further theoretical developments and applications to simulation. Branching process models for electron-photon cascades and related physical processes will be studied with additional applications of large deviation methods. Theoretical developments for general stochastic control problems will include approximation methodology and existence theorems for optimal singular controls. New types of stochastic equations for Markov processes will be studied. Markov processes are random processes whose future behavior depends on the past only through the present. If you know the current "state" of a system having the Markov property, your ability to predict the future is not helped by additional information about the past. Markov processes play an important role in the study of population genetics and in descriptions of changing populations more generally, and will be studied in this context. Markov processes are also important as models of physical phenomena such as cosmic rays, which will be investigated. Problems to be considered include control, approximation, and extreme behavior. An additional direction of research involves processes which fail to have the "Markov property", but which have special kinds of long range memory.