Communications in Mathematical Sciences

Volume 18 (2020)

Number 7

Anomalous diffusion in comb-shaped domains and graphs

Pages: 1815 – 1862

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n7.a2

Authors

Samuel Cohn (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Gautam Iyer (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

James Nolen (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Robert L. Pego (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Abstract

In this paper we study the asymptotic behavior of Brownian motion in both comb-shaped planar domains, and comb-shaped graphs. We show convergence to a limiting process when both the spacing between the teeth and the width of the teeth vanish at the same rate. The limiting process exhibits an anomalous diffusive behavior and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. In the two dimensional setting the main technical step is an oscillation estimate for a Neumann problem, which we prove here using a probabilistic argument. In the one dimensional setting we provide both a direct SDE proof, and a proof using the trapped Brownian motion framework in Ben Arous et al. (Ann. Probab. ’15).

Keywords

anomalous diffusion, sticky Brownian motion, fractional kinetic process, comb models

2010 Mathematics Subject Classification

35B27, 60G22

Full Text (PDF format)

This work was partially supported by the National Science Foundation under grants DMS-1252912, DMS-1351653, DMS-1515400, DMS-1814147, and the Center for Nonlinear Analysis.

Received 23 September 2019

Accepted 1 April 2020

Published 11 December 2020