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

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

Research output: Contribution to journalArticle

5 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)897-955
Number of pages59
JournalAnnals of Probability
Volume46
Issue number2
DOIs
StatePublished - Mar 1 2018

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Time Scales
Fractional
Kinetics
Brownian motion
Averaging Principle
Advection-diffusion Equation
Random Perturbation
Anomalous Diffusion
Limiting Behavior
Local Time
Small Perturbations
Homogenization
Trajectory
Time scales
Cell
Theorem
Modeling

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Hairer, Martin ; Iyern, Gautam ; Koralov, Leonid ; Novikov, Alexei ; Pajor-Gyulai, Zsolt. / A fractional kinetic process describing the intermediate time behaviour of cellular flows. In: Annals of Probability. 2018 ; Vol. 46, No. 2. pp. 897-955.
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A fractional kinetic process describing the intermediate time behaviour of cellular flows. / Hairer, Martin; Iyern, Gautam; Koralov, Leonid; Novikov, Alexei; Pajor-Gyulai, Zsolt.

In: Annals of Probability, Vol. 46, No. 2, 01.03.2018, p. 897-955.

Research output: Contribution to journalArticle

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