This paper is concerned with accurate asymptotic estimates for exit time probabilities associated with nearly deterministic Markov diffusions. The exit time probabilities are expressed as asymptotic series of WKB type in a small parameter, which measures the strength of the random Brownian motion inputs. This series is valid in certain regions in which the minimum action function $u(x,s)$ is a smooth function of state $x$ and time $s$. The function $u$ is a solution to the corresponding Hamilton-Jacobi PDE of first order.
"Asymptotic Series and Exit Time Probabilities." Ann. Probab. 20 (3) 1369 - 1384, July, 1992. https://doi.org/10.1214/aop/1176989695