For a smooth vector field in a neighborhood of a critical point with all positive eigenvalues of the linearization, we consider the associated dynamics perturbed by white noise. Using Malliavin calculus tools, we obtain polynomial asymptotics for probabilities of atypically long exit times in the vanishing noise limit.
"Long exit times near a repelling equilibrium." Ann. Appl. Probab. 31 (2) 594 - 624, April 2021. https://doi.org/10.1214/20-AAP1599