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 journalArticlepeer-review

9 Scopus citations

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

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint Dive into the research topics of 'A fractional kinetic process describing the intermediate time behaviour of cellular flows'. Together they form a unique fingerprint.

Cite this