Large Deviations and rare events in diffusion processes

This thesis collect some of the main results in the theory of Large Deviations for diffusion process. The material in the first two chapters comes from the classical book by Freidlin and Wentzell and contains the main results, such as Schilder's theorem and its extension to diffusion processes. As an application of these results, we consider in the second chapter the problem of the exit from a stable domain. In dealing with this problem we introduce the notion of Instanton and Quasi-potential. Since these objects, and the associated minimization problem, are central in the theory of Large Deviations, the third chapter deals with the problem of explicitly computing them, introducing for example, the notion of geometric action, a parametrization-free rewrite of the classical Freidlin-Wentzell action functional, which is well suited for computational methods. In the last chapter we illustrate a simple example of numerical computations connected with the study of rare and "extreme" events in dynamical system with a small noise. We compare the numerical results with Monte-Carlo simulations to demonstrate the possible predictive power of the theory of large deviations in such contexts