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March 2018 A fractional kinetic process describing the intermediate time behaviour of cellular flows
Martin Hairer, Gautam Iyer, Leonid Koralov, Alexei Novikov, Zsolt Pajor-Gyulai
Ann. Probab. 46(2): 897-955 (March 2018). DOI: 10.1214/17-AOP1196

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.

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Martin Hairer. Gautam Iyer. Leonid Koralov. Alexei Novikov. Zsolt Pajor-Gyulai. "A fractional kinetic process describing the intermediate time behaviour of cellular flows." Ann. Probab. 46 (2) 897 - 955, March 2018. https://doi.org/10.1214/17-AOP1196

Information

Received: 1 July 2016; Revised: 1 April 2017; Published: March 2018
First available in Project Euclid: 9 March 2018

zbMATH: 06864076
MathSciNet: MR3773377
Digital Object Identifier: 10.1214/17-AOP1196

Subjects:
Primary: 26A33, 35R11, 60F17, 60H10, 60H30, 76R50

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 2 • March 2018
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