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

Thomas Vojta, Alex Warhover

Published 2020-12-05Version 1

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. For both stochastic processes and both geometries, 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. The probability density of the fractional Langevin equation can be mapped onto that of fractional Brownian motion driven by the same noise by replacing $\alpha$ with $2-\alpha$. We also contrast the behavior near absorbing walls with the strong accumulation and depletion effects recently observed for fractional Brownian motion and the fractional Langevin equation near reflecting boundaries.