  We derive the probability density function of the positive occupation time of
one-dimensional Brownian motion with two-valued drift. Long time asymptotics of
the density are also computed. We use the result to describe the transitional
probability density function of a general N-dimensional system of stochastic
differential equations representing stochastically perturbed sliding motion of
a discontinuous, piecewise-smooth vector field on short time frames. A
description of the density at larger times is obtained via an asymptotic
expansion of the Fokker-Planck equation.

Probability