Fractional Brownian motion, a stochastic process with long-time correlations between its increments, is a prototypical model for anomalous diffusion. We analyze fractional Brownian motion in the presence of a reflecting wall by means of Monte Carlo simulations. Whereas the mean-square displacement of the particle shows the expected anomalous diffusion behavior (x2) ~ tα, the interplay between the geometric confinement and the long-time memory leads to a highly non-Gaussian probability density function with a power-law singularity at the barrier. In the superdiffusive case α > 1, the particles accumulate at the barrier leading to a divergence of the probability density. For subdiffusion α < 1, in contrast, the probability density is depleted close to the barrier. We discuss implications of these findings, in particular, for applications that are dominated by rare events.



Research Center/Lab(s)

Center for High Performance Computing Research


This work was supported by the NSF under Grants No. PHY-1125915 and No. DMR- 1506152 and by the Sao Paulo Research Foundation (FAPESP) under Grant No. 2017/08631-0. T.V. is grateful for the hospitality of the Kavli Institute for Theoretical Physics, Santa Barbara where part of the research was performed.

Keywords and Phrases

Diffusion; Diffusion barriers; Intelligent systems; Monte Carlo methods; Probability; Probability density function; Random processes; Stochastic models; Stochastic systems, Anomalous diffusion; Fractional brownian motion; Geometric confinement; Long-time correlations; Mean square displacement; Non-gaussian probability density function; Power-law singularity; Probability densities, Brownian movement

International Standard Serial Number (ISSN)

2470-0045: 2470-0053

Document Type

Article - Journal

Document Version

Final Version

File Type





© 2018 American Physical Society (APS), All rights reserved.

Publication Date

01 Feb 2018

Included in

Physics Commons