Analytic crossing probabilities for certain barriers by Brownian motion

We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval [0, T] via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class of barriers on [0, T] by proving a Schwartz distribution version of the method of images. Analytic expressions for crossing probabilities and related densities are given for new explicit and semi-explicit barriers.