  The objective of this dissertation is to prove a scaling limit for the exit
of a domain problem of a small noise system with underlying hyperbolic
dynamics. In this case, Large Deviation kind of estimates fail to provide a
complete picture of the dynamics of the system under consideration. This is so
because the rate function given by the Large Deviation Principle has several
minimizing trajectories hence making them indistinguishable at the exponential
level. We propose a pathwise approach based on the theory of normal forms
combined with geometrical arguments. We prove a scaling limits and provide the
state of the art results in related problems.
