Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls

TL;DR

This study uses large-scale simulations to analyze the probability density functions of fractional Brownian motion and the fractional Langevin equation with absorbing walls, revealing a mapping between their properties and confirming conjectures about their near-wall behavior.

Contribution

The paper demonstrates a mapping between fractional Langevin equation and fractional Brownian motion properties with a specific exponent substitution, and provides detailed analysis of probability densities near absorbing walls.

Findings

Probability density near an absorbing wall behaves as P(x) ~ x^kappa.

For fractional Brownian motion, kappa = 2/alpha - 1.

Simulation results support the conjectured behavior of the probability density.

Abstract

Fractional Brownian motion and the fractional Langevin equation are models of anomalous diffusion processes characterized by long-range power-law correlations in time. We employ large-scale computer simulations to study these models in two geometries, (i) the spreading of particles on a semi-infinite domain with an absorbing wall at one end and (ii) the stationary state on a finite interval with absorbing boundaries at both ends and a source in the center. We demonstrate that the probability density and other properties of the fractional Langevin equation can be mapped onto the corresponding quantities of fractional Brownian motion driven by the same noise if the anomalous diffusion exponent $\alpha$ is replaced by $2-\alpha$. In contrast, the properties of fractional Brownian motion and the fractional Langevin equation with reflecting boundaries were recently shown to differ from each other qualitatively. Specifically, we find that the probability density close to an absorbing wall behaves as $P(x) \sim x^\kappa$ with the distance $x$ from the wall in the long-time limit. In the case of fractional Brownian motion, $\kappa$ varies with the anomalous diffusion exponent $\alpha$ as $\kappa=2/\alpha -1$, as was conjectured previously. We also compare our simulation results to a perturbative analytical approach to fractional Brownian motion.