Student Probability/PDE Seminar: A Fractional Kinetic Process Describing the Intermediate Time Behaviour of Cellular Flows

Seminar | March 1 | 2:10-3:30 p.m. | 891 Evans Hall

 Alexei Novikov, Penn State University

 Department of Mathematics

This is joint work with Martin Hairer, Gautam Iyer, Leonid Koralov, and Zsolt Pajor-Gyulai. This work 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.