Athreya, DwijavantiWe characterize the phenomenon of metastability for a small random perturbation of a nearly-Hamiltonian dynamical system with one degree of freedom. We use the averaging principle and the theory of large deviations to prove that a metastable state is, in general, not a single state but rather a nondegenerate probability measure across the stable equilibrium points of the unperturbed Hamiltonian system. The set of all possible ``metastable distributions" is a finite set that is independent of the stochastic perturbation. These results lead to a generalization of metastability for systems close to Hamiltonian ones.en-USDissertationMathematicsAveragingLarge deviationsMetastability