Fractional Brownian motion with a reflecting wall

TL;DR

This paper investigates fractional Brownian motion near a reflecting wall, revealing how long-time correlations and confinement produce non-Gaussian distributions with singularities, affecting particle accumulation or depletion depending on the diffusion type.

Contribution

It provides the first detailed analysis of fractional Brownian motion with a reflecting boundary, highlighting the impact of long-range correlations on the probability density function.

Findings

Superdiffusive particles accumulate at the barrier, causing divergence in probability density.

Subdiffusive particles are depleted near the barrier, reducing probability density.

The interplay between confinement and memory effects leads to non-Gaussian, power-law singularities.

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 $\langle x^2 \rangle \sim t^\alpha$, 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, $\alpha> 1$, the particles accumulate at the barrier leading to a divergence of the probability density. For subdiffusion, $\alpha < 1$, 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.