A FRACTIONAL KINETIC PROCESS DESCRIBING THE INTERMEDIATE TIME BEHAVIOUR OF CELLULAR FLOWS
File(s)
Accepted version
Author(s)
Hairer, Martin
Iyer, Gautam
Koralov, Leonid
Novikov, Alexei
Pajor-Gyulai, Zsolt
Type
Journal Article
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.
Date Issued
2018-03-01
Date Acceptance
2017-04-27
Citation
ANNALS OF PROBABILITY, 2018, 46 (2), pp.897-955
ISSN
0091-1798
Publisher
INST MATHEMATICAL STATISTICS
Start Page
897
End Page
955
Journal / Book Title
ANNALS OF PROBABILITY
Volume
46
Issue
2
Copyright Statement
© The Authors
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000430922600006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
Fractional kinetics
cellular flows
averaging principle
homogenization
ANOMALOUS DIFFUSION
RANDOM PERTURBATIONS
HAMILTONIAN FLOWS
PERIODIC FLOWS
RANDOM-WALKS
HOMOGENIZATION
PRINCIPLE
SYSTEMS
GRAPHS
SCALES
Notes
47 pages
Publication Status
Published