Anomalous diffusion in comb-shaped domains and graphs

TL;DR

This paper investigates the asymptotic behavior of Brownian motion in comb-shaped domains and graphs, revealing a limiting anomalous diffusive process characterized by a time-changed Brownian motion influenced by local time.

Contribution

It introduces a new limit process for Brownian motion in comb-shaped structures as the teeth vanish, combining probabilistic and PDE techniques to analyze the convergence.

Findings

Convergence to a time-changed Brownian motion with local time.

The limiting process exhibits anomalous diffusion behavior.

Established oscillation estimates for Neumann problems using probabilistic methods.

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 \emph{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 \etal (Ann.\ Probab.\ '15).