Abstract

Rare events, metastability and Monte Carlo methods for stochastic dynamical systems have been of central scientific interest for many years now. In this article we focus on rough energy landscapes, that are modeled as multiscale stochastic dynamical systems perturbed by small noise. Large deviations deals with the estimation of rare events. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. We describe how to design asymptotically provably efficient importance sampling schemes for the estimation of associated rare event probabilities, such as exit probabilities,hitting probabilities, hitting times, and expectations of functionals of interest. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. Theoretical results are supplemented by numerical simulation studies.