  We perform a time-dependent study of the driven dynamics of overdamped
particles which are placed in a one-dimensional, piecewise linear random
potential. This set-up of spatially quenched disorder then exerts a dichotomous
varying random force on the particles. We derive the path integral
representation of the resulting probability density function for the position
of the particles and transform this quantity of interest into the form of a
Fourier integral. In doing so, the evolution of the probability density can be
investigated analytically for finite times. It is demonstrated that the
probability density contains both a $\delta$-singular contribution and a
regular part. While the former part plays a dominant role at short times, the
latter rules the behavior at large evolution times. The slow approach of the
probability density to a limiting Gaussian form as time tends to infinity is
elucidated in detail.
