  Classical Stokes' drift is the small time-averaged drift velocity of
suspended non-diffusing particles in a fluid due to the presence of a wave. We
consider the effect of adding diffusion to the motion of the particles, and
show in particular that a non-zero time-averaged drift velocity exists in
general even when the classical Stokes' drift is zero. Our results are obtained
from a general procedure for calculating ensemble-averaged Lagrangian mean
velocities for motion that is close to Brownian, and are verified by numerical
simulations in the case of sinusoidal forcing.
