Fractional Lengevin equation with reflecting barrier

The Fractional Langevin equation describes a the motion of a particle under the influence of a random force with long-time correlations. This stochastic differential equation is a common model for anomalous diffusion. We investigate the fractional Langevin equation in the presence of a reflecting wall using Monte Carlo simulations. The mean-square displacement shows the expected anomalous diffusion behavior, 〈x^2〉 ∼ t^2-α , as in the unconfined case. However, the probability density close to the wall shows highly non-Gaussian behavior. For reference, we compare our results to reflected fractional Brownian motion for which the probability density shows a power law singularity at the barrier [1].

[1] A.H.O. Wada and T. Vojta, Physics Review E 97, 0201012 (2018)