  We investigate the average velocity of Brownian particles driven by a
constant external force when constrained to move in two-dimensional,
weakly-corrugated channels. We consider both the geometric confinement of the
particles between solid walls as well as the soft confinement induced by a
periodic potential. Using perturbation methods we show that the leading order
correction to the marginal probability distribution of particles in the case of
soft confinement is equal to that obtained in the case of geometric
confinement, provided that the (configuration) integral over the cross-section
of the confining potential is equal to the width of the solid channel. We then
calculate the probability distribution and average velocity in the case of a
sinusoidal variation in the width of the channels. The reduction on the average
velocity is larger in the case of soft channels at small P\'eclet numbers and
for relatively narrow channels and the opposite is true at large P\'eclet
numbers and for wide channels. In the limit of large P\'eclet numbers the
convergence to bulk velocity is faster in the case of soft channels. The
leading order correction to the average velocity and marginal probability
distribution agree well with Brownian Dynamics simulations for the two types of
confinement and over a wide range of P\'eclet numbers.
