Publication 18-CNA-018
Anomalous Diffusion In One And Two Dimensional Combs
Samuel Cohn
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
samuelcohn032@gmail.com
Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu
James Nolen
Department of Mathematics
Duke University
243 Physics Building, Durham, NC 27708
nolen@math.duke.edu
Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@andrew.cmu.edu
Abstract: In this paper we study the asymptotic behavior of Brownian motion in both one and two dimensional comb-like domains. We show convergence to a limiting process when both the spacing between the teeth and the probability of entering a tooth vanish at the same rate. The limiting process may exhibit
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.
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