  The problem of Brownian motion in a periodic potential, under the influence
of external forcing, which is either random or periodic in time, is studied in
this paper. Multiscale techniques are used to derive general formulae for the
steady state particle current and the effective diffusion tensor. These
formulae are then applied to calculate the effective diffusion coefficient for
a Brownian particle in a periodic potential driven simultaneously by additive
Gaussian white and colored noise. Our theoretical findings are supported by
numerical simulations.
