A FRACTIONAL KINETIC PROCESS DESCRIBING THE INTERMEDIATE TIME BEHAVIOUR OF CELLULAR FLOWS

• 成果类型: Article
• 署名作者: Hairer, Martin; Iyer, Gautam; Koralov, Leonid; Novikov, Alexei; Pajor-Gyulai, Zsolt
• 署名单位: University of Warwick; Carnegie Mellon University; University System of Maryland; University of Maryland College Park; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park; New York University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1196
• 发表日期: 2018
• 页码: 897-955
• 关键词: random perturbations; anomalous diffusion; homogenization; graphs

摘要

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.