  We discuss the dynamics of a Brownian particle under the influence of a
spatially periodic noise strength in one dimension using analytical theory and
computer simulations. In the absence of a deterministic force, the Langevin
equation can be integrated formally exactly. We determine the short- and
long-time behaviour of the mean displacement (MD) and mean-squared displacement
(MSD). In particular we find a very slow dynamics for the mean displacement,
scaling as $t^{-1/2}$ with time $t$. Placed under an additional external
periodic force near the critical tilt value we compute the stationary current
obtained from the corresponding Fokker-Planck equation and identify an
essential singularity if the minimum of the noise strength is zero. Finally, in
order to further elucidate the effect of the random periodic driving on the
diffusion process, we introduce a phase factor in the spatial noise with
respect to the external periodic force and identify the value of the phase
shift for which the random force exerts its strongest effect on the long-time
drift velocity and diffusion coefficient.
