Non-Gaussian behavior of reflected fractional Brownian motion

Abstract

A possible mechanism leading to anomalous diffusion is the presence of long-range correlations in time between the displacements of the particles. Fractional Brownian motion, a non-Markovian self-similar Gaussian process with stationary increments, is a prototypical model for this situation. Here, we extend the previous results found for unbiased reflected fractional Brownian motion [Phys. Rev. E 97, 020102(R) (2018)] to the biased case by means of Monte Carlo simulations and scaling arguments. We demonstrate that the interplay between the reflecting wall and the correlations leads to highly non-Gaussian probability densities of the particle position $x$ close to the reflecting wall. Specifically, the probability density $P(x)$ develops a power-law singularity $P \sim x^\kappa$ with $\kappa < 0$ if the correlations are positive (persistent) and $\kappa > 0$ if the correlations are negative (antipersistent). We also analyze the behavior of the large-$x$ tail of the stationary probability density reached for bias towards the wall, the average displacements of the walker, and the first-passage time, i.e., the time it takes for the walker reach position $x$ for the first time.