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Publication 16-CNA-016

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

Martin Hairer
University of Warwick

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Leonid Koralov
University of Maryland

Alexei Novikov
Department of Mathematics
Pennsylvania State University
State College PA 16802

Zsolt Pajor-Gyulai
New York University

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.

Get the paper in its entirety as  16-CNA-016.pdf

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