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

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 α is replaced by 2 − α. 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) ∼ x κ with the distance x from the wall in the long-time limit. In the case of fractional Brownian motion, κ varies with the anomalous diffusion exponent α as κ = 2/α − 1, as was conjectured previously. We also compare our simulation results to a perturbative analytical approach to fractional Brownian motion.

Tempered fractional Brownian motion on finite intervals

“…We also note that there is an interesting similarity between the behavior of the probability density close to a reflecting wall (at position w), P ∼ |x − w| 2/α−2 , and the corresponding behavior close to an absorbing wall, P ∼ |x − w| 2/α−1 [43,44,48].…”

“…In addition, properties of FBM close to an absorbing boundary were investigated in Refs. [42][43][44][45][46][47][48].…”

A note on Euler method for the overdamped generalized Langevin equation with fractional noise

“…It is meaningful and indispensable to use the numerical method to generate the approximation of the density function of {x(t)} t∈[0,T ] . In numerical aspect, we are only aware that [19,20] use the Euler-Maruyama (EM) method to simulate the density function of the solution of underdamped GLEs with reflecting and absorbing walls, respectively. The numerical research on the density function for Eq.…”