For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.
"Malliavin calculus approach to long exit times from an unstable equilibrium." Ann. Appl. Probab. 29 (2) 827 - 850, April 2019. https://doi.org/10.1214/18-AAP1387