  This paper studies the intermediate time behaviour of a small random
perturbation of a periodic cellular flow. Our main result shows that on time
scales shorter than the diffusive time scale, the limiting behaviour of
trajectories that start close enough to cell boundaries is a fractional kinetic
process: A Brownian motion time changed by the local time of an independent
Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key
step is to establish an analogous averaging principle on shorter time scales.
As a consequence of our main theorem, we obtain a homogenization result for the
associated advection-diffusion equation. We show that on intermediate time
scales the effective equation is a fractional time PDE that arises in modelling
anomalous diffusion.
