Mathematical Sciences: Topics in Large Deviations

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9627045 de Acosta The investigator proposes to contribute to the study of basic questions in Large Deviations related to some important stochastic sequences or ppocesses. This includes the development of suitable abstract mathematical tools, the identification of models which lead naturally to interesting large deviation questions, and the computation of the analytical expression of specific rate functions. The scope of the project includes the following situations: (i) trajectories of Markov processes (ii) trajectories of certain classes of semimartingales (iii) empirical measures under various dependence conditions. A fundamental theme in Probability Theory and its applications is the study of situations in which a sequence of averages constructed from a stochastic sequence converges in some sense to a deterministic limit. Large Deviation theory deals with the problem of estimating the small probabilities that the averages will deviate from the "typical behavior" represented by the limit. Here are two situations to which the proposed research has direct or potential relevance; (a) Probabilistic models are increasingly used in analyzing the performance of complex communication systems and computer networks. Large Deviation theory (in particular, large deviations of certain Markov processes) is a powerful tool in the analysis of rare events (that is, deviations from "normal behavior") in large systems of those types. (b) Large Deviation theory is very useful in the analysis of rare events associated to thermodynamic limits in several important models in Statistical Mechanics.