Metastability for a Class of Dynamical Systems Subject to Small Random Perturbations

Abstract

We consider dynamical systems in $\mathbb{R}^d$ driven by a vector field $b(x) = - \nabla a(x)$, where $a$ is a double-well potential with some smoothness conditions. We show that these dynamical systems when subjected to a small random disturbance exhibit metastable behavior in the sense defined in [2]. More precisely, we prove that the process of moving averages along a path of such a system converges in law when properly normalized to a jump Markov process. The main tool for our analysis is the theory of Freidlin and Wentzell [7].