  We analyze the mean squared displacement of a Brownian particle in a medium
with a spatially varying local diffusivity which is assumed to be periodic.
When the system is asymptotically diffusive the mean squared displacement,
characterizing the dispersion in the system, is, at late times, a linear
function of time. A Kubo type formula is given for the mean squared
displacement which allows the recovery of some known results for the effective
diffusion constant $D_e$ in a direct way, but also allows an understanding of
the asymptotic approach to the diffusive limit. In particular, as well as as
computing the slope of a linear fit to the late time mean squared displacement,
we find a formula for the constant where the fit intersects the y axis.
