  We investigate piecewise-linear stochastic models as with regards to the
probability distribution of functionals of the stochastic processes, a question
which occurs frequently in large deviation theory. The functionals that we are
looking into in detail are related to the time a stochastic process spends at a
phase space point or in a phase space region, as well as to the motion with
inertia. For a Langevin equation with discontinuous drift, we extend the
so-called backward Fokker-Planck technique for nonnegative support functionals
to arbitrary support functionals, to derive explicit expressions for the
moments of the functional. Explicit solutions for the moments and for the
distribution of the so-called local time, the occupation time and the
displacement are derived for the Brownian motion with dry friction, including
quantitative measures to characterize deviation from Gaussian behaviour in the
asymptotic long time limit.
