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This paper studies the intermediate time behaviour of a small randomperturbation of a periodic cellular flow. Our main result shows that on timescales shorter than the diffusive time scale, the limiting behaviour oftrajectories that start close enough to cell boundaries is a fractional kineticprocess: A Brownian motion time changed by the local time of an independentBrownian motion. Our proof uses the Freidlin-Wentzell framework, and the keystep is to establish an analogous averaging principle on shorter time scales.As a consequence of our main theorem, we obtain a homogenization result for theassociated advection-diffusion equation. We show that on intermediate timescales the effective equation is a fractional time PDE that arises in modellinganomalous diffusion.

A fractional kinetic process describing the intermediate time behaviour of cellular flows