Markov Processes

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In the original work of Esscher and Cramer Large Deviations arise through what is now known as Esscher or Cramer tilt, that changes the underlying distribution so that what was originally a rare event is now a very likely event. The exact tilt that leads to the large deviation allows one to calculate the precise rate of the large deviation. This provides a clue as to what the conditional distribution is, given that a particular rare event has occurred. While the original work is in the context of sums of independent random variables, subsequent work on large deviations has extended this idea considerably. It is the aim of this proposal to investigate this in the context of a random walk in a random environment. In particular if the walk travels with an unlikely velocity, what would it have experienced?Large deviations deals with estimating probabilities of rare events. Rare events do occur and the theory deals with determining exactly how rare they are. When a rare event occurs it is not isolated. Other rare events happen as well. The same phenomenon that generated the rare event could very well have spawned other rare events. The theory deals with predicting such other rare events.One starts with a world of models as well as an assumption about a specific model. An event with very small probability under this model occurs. This changes one's belief in the particular model and a new model is chosen that is consistent with the rare event.If there are many such models an optimal one is chosen in some way. This model makes predictions, which were perhaps rare under the old model but not any more.