Fractional Brownian motion with a reflecting wall

Abstract

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. While the mean-square displacement of the particle shows the expected anomalous diffusion behavior $\langle x^2 \rangle \sim t^α$, 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.