  We provide an analytic solution to the first-passage time (FPT) problem of a
piecewise-smooth stochastic model, namely Brownian motion with dry friction,
using two different but closely related approaches which are based on
eigenfunction decompositions on the one hand and on the backward Kolmogorov
equation on the other. For the simple case containing only dry friction, a
phase transition phenomenon in the spectrum is found which relates to the
position of the exit point, and which affects the tail of the FPT distribution.
For the model containing as well a driving force and viscous friction the
impact of the corresponding stick-slip transition and of the transition to
ballistic exit is evaluated quantitatively. The proposed model is one of the
very few cases where FPT properties are accessible by analytical means.
