Rare event simulation in the neighborhood of a rest point

In this paper, we construct efficient importance sampling Monte Carlo schemes for finite time exit probabilities in the presence of rest points. We focus on reversible diffusion processes with small noise that have an asymptotically stable equilibrium point. The main novelty of the work is the inclusion of rest points in the domain of interest. We motivate the construction of schemes that perform well both asymptotically and non-asymptotically. We concentrate on the regime where the noise is small and the time horizon is large. Examples and simulation results are provided.