  In this paper the author studies the problem of the homogenization of a
diffusion perturbed by a periodic reflection invariant vector field. The vector
field is assumed to have fixed direction but varying amplitude. The existence
of a homogenized limit is proven and formulas for the effective diffusion
constant are given. In dimension $d=1$ the effective diffusion constant is
always less than the constant for the pure diffusion. In $d>1$ this property no
longer holds in general.
