We show that under mild conditions the hitting time distributions of a general stochastic process are the unique solutions of an abstract Dirichlet problem.
Dudley Paul Johnson. "Hitting Time Distributions for General Stochastic Processes." Ann. Math. Statist. 43 (5) 1716 - 1718, October, 1972. https://doi.org/10.1214/aoms/1177692408