Markov Processes

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This proposal is partly about large deviations in different contexts. In studying the scaling limits of interacting particle limits, one usually proves laws of large numbers. They arise in different contexts and have corresponding large deviation results that are related. Random walks and diffusions in a random environment are also expected, under suitable conditions, to satisfy law of large numbers, central limit theorem and large deviations. The large deviation theory has close connection to the theory of homogenization of random Hamilton-Jacobi-Bellman equations with vanishing viscosity. We expect to continue with our investigation in these areas. In complex interactive systems, noise often acts as a stabilizing force and guides the system along a predictable deterministic path. Deviations from this behavior are rare. One goal of this project is to quantify and measure how rare a given deviation is and if such a deviation should occur what is most likely to have caused it?