Markov Processes

在Esscher和Cramer的原始工作中，大偏差是通过现在被称为Esscher或Cramer倾斜的东西出现的，它改变了潜在的分布，因此最初是罕见的事件现在是非常可能的事件。导致大偏差的精确倾斜允许计算大偏差的精确速率。这提供了一个线索，告诉我们条件分布是什么，假设一个特定的罕见事件已经发生。虽然最初的工作是在独立随机变量的总和的背景下，随后的工作大偏差大大扩展了这一想法。这是本建议的目的，调查这在随机环境中的随机游走的背景下。特别是，如果步行以一个不太可能的速度行进，它会经历什么？大偏差处理估计罕见事件的概率。 稀有事件确实会发生，而这个理论就是要确定它们到底有多稀有。当一个罕见的事件发生时，它不是孤立的。其他罕见的事件也会发生。产生罕见事件的同一现象很可能产生其他罕见事件。该理论处理的是对其他罕见事件的预测，一开始是一个模型世界，以及对一个特定模型的假设。在此模型下发生概率非常小的事件。这改变了人们对特定模型的信念，并选择了一个与罕见事件一致的新模型，如果有许多这样的模型，则以某种方式选择一个最佳模型。 这个模型可以做出预测，这在旧模型下可能是罕见的，但现在已经不是了。

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.