  In this paper we study the unidirectional transport effect for Brownian
ratchets modeled by Fokker-Planck-type equations. In particular, we consider
the adiabatic and semiadiabatic limits for tilting ratchets, generic ratchets
with small diffusion, and the multi-state chemical ratchets. Having established
a linear relation between the bulk transport velocity and the bi-periodic
solution, and using relative entropy estimates and new functional inequalities,
we obtain explicit asymptotic formulas for the transport velocity and
qualitative results concerning the direction of transport. In particular, we
prove the conjecture by Blanchet, Dolbeault and Kowalczyk that the bulk
velocity of the stochastic Stokes' drift is non-zero for every non-constant
potential.
