Escaping from an attractor: Importance sampling and rest points I

TL;DR

This paper develops importance sampling methods for estimating escape probabilities in small noise diffusions, effectively handling rest points and ensuring good performance across different noise scales and time horizons.

Contribution

It introduces importance sampling schemes that are provably efficient both for fixed noise levels and asymptotically, even with rest points in the domain.

Findings

Schemes perform well for fixed noise sizes.

Methods remain effective as noise diminishes.

Simulation results confirm theoretical guarantees.

Abstract

We discuss importance sampling schemes for the estimation of finite time exit probabilities of small noise diffusions that involve escape from an equilibrium. A factor that complicates the analysis is that rest points are included in the domain of interest. We build importance sampling schemes with provably good performance both pre-asymptotically, that is, for fixed size of the noise, and asymptotically, that is, as the size of the noise goes to zero, and that do not degrade as the time horizon gets large. Simulation studies demonstrate the theoretical results.