The Annals of Probability

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

Martin Hairer, Gautam Iyer, Leonid Koralov, Alexei Novikov, and Zsolt Pajor-Gyulai

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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.

Article information

Source
Ann. Probab., Volume 46, Number 2 (2018), 897-955.

Dates
Received: July 2016
Revised: April 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1520586272

Digital Object Identifier
doi:10.1214/17-AOP1196

Mathematical Reviews number (MathSciNet)
MR3773377

Zentralblatt MATH identifier
06864076

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 60F17: Functional limit theorems; invariance principles 26A33: Fractional derivatives and integrals 35R11: Fractional partial differential equations 76R50: Diffusion [See also 60J60]

Keywords
Fractional kinetics cellular flows averaging principle homogenization

Citation

Hairer, Martin; Iyer, Gautam; Koralov, Leonid; Novikov, Alexei; Pajor-Gyulai, Zsolt. A fractional kinetic process describing the intermediate time behaviour of cellular flows. Ann. Probab. 46 (2018), no. 2, 897--955. doi:10.1214/17-AOP1196. https://projecteuclid.org/euclid.aop/1520586272


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