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

TL;DR

This paper analyzes the intermediate time behavior of cellular flows under small random perturbations, revealing a fractional kinetic process characterized by a time-changed Brownian motion, and establishes related homogenization results.

Contribution

It introduces a fractional kinetic process model for cellular flows at intermediate times and proves an averaging principle on shorter time scales using the Freidlin-Wentzell framework.

Findings

Trajectories near cell boundaries follow a fractional kinetic process.

Homogenization yields a fractional time PDE for anomalous diffusion.

Intermediate time scales exhibit non-classical diffusion behavior.

Abstract

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.