  We investigate the validity and accuracy of weak-noise (saddle-point or
instanton) approximations for piecewise-smooth stochastic differential
equations (SDEs), taking as an illustrative example a piecewise-constant SDE,
which serves as a simple model of Brownian motion with solid friction. For this
model, we show that the weak-noise approximation of the path integral correctly
reproduces the known propagator of the SDE at lowest order in the noise power,
as well as the main features of the exact propagator with higher-order
corrections, provided that the singularity of the path integral associated with
the non-smooth SDE is treated with some heuristics. We also show that, as in
the case of smooth SDEs, the deterministic paths of the noiseless system
correctly describe the behaviour of the non-smooth SDE in the low-noise limit.
Finally, we consider a smooth regularisation of the piecewise-constant SDE and
study to which extent this regularisation can rectify some of the problems
encountered when dealing with discontinuous drifts and singularities in SDEs.
