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 disp
ersion 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.
