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. We also discuss our program on rare transitions in noisy heteroclinic networks.
- Keywords: vanishing noise limit
- exit problem
- polynomial decay
- unstable equilibrium
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty