Tails of exit times from unstable equilibria on the line

TL;DR

This paper provides a simple proof for the asymptotic behavior of exit times from unstable equilibria in noisy one-dimensional systems, detailing tail distributions and exit times under small noise perturbations.

Contribution

It offers an elementary proof of known results on exit time asymptotics, avoiding complex Malliavin calculus techniques.

Findings

Precise asymptotics for tail of exit times

Distribution conditioned on long exits characterized

Simplified proof of existing results

Abstract

For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.