Publication 16-CNA-016

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

Martin Hairer
University of Warwick
M.Hairer@Warwick.ac.uk

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Leonid Koralov
University of Maryland
koralov@math.umd.edu

Alexei Novikov
Department of Mathematics
Pennsylvania State University
State College PA 16802
anovikov@math.psu.edu

Zsolt Pajor-Gyulai
New York University
zsolt@cims.nyu.edu

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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