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.




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.

Included in

Physics Commons