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Large Deviations for Random Walks in Random Environments (LargeDevRWRE)
Start date: Sep 1, 2012, End date: Aug 31, 2016 PROJECT  FINISHED 

Random walk in random environment (RWRE) is one of the standard models in the study of random media. Originally motivated by molecular biology applications and subsequently used in the modeling of disordered systems, it has been a constant source of fascinating mathematical phenomena and challenging open problems over the last four decades. The theory of large deviations is a cornerstone of modern probability, with applications to statistical mechanics, information theory, control engineering, risk management and many other fields.The main goal of the proposed project is to attain a clear and detailed understanding of the large deviation properties of multidimensional RWRE. The objectives can be summarized as follows: (i) Obtain simple expressions for the quenched and the averaged large deviation rate functions. (ii) Compare these rate functions and find a formula that links them. (iii) Identify the dynamics of the walk when it is conditioned to have an atypical velocity. (iv) Characterize when the new dynamics is obtained from the original one by a simple change of measure. (v) Find the ergodic invariant measure on environments from the point of view of the particle under such a conditioning.The project will build on the past accomplishments of the researcher on this topic and will involve variational formulas for the rate functions and their minimizers. The researcher will start by studying directed RWRE and focus on specific environment distributions. He will then consider general distributions and adapt the results to ballistic undirected walks by making use of regeneration times. The techniques are robust, they also cover polymers in random environments, and are likely to shed light on the open problem of strong vs. very strong disorder. The project has applications to some of the most active areas of probability theory, statistical mechanics, mathematical physics, and stochastic PDEs.
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