Tempered fractional Brownian motion on finite intervals

Abstract

Diffusive transport in many complex systems features a crossover between anomalous diffusion at short times and normal diffusion at long times. This behavior can be mathematically modeled by cutting off (tempering) beyond a mesoscopic correlation time the power-law correlations between the increments of fractional Brownian motion. Here, we investigate such tempered fractional Brownian motion confined to a finite interval by reflecting walls. Specifically, we explore how the tempering of the long-time correlations affects the strong accumulation and depletion of particles near reflecting boundaries recently discovered for untempered fractional Brownian motion. We find that exponential tempering introduces a characteristic size for the accumulation and depletion zones but does not affect the functional form of the probability density close to the wall. In contrast, power-law tempering leads to more complex behavior that differs between the superdiffusive and subdiffusive cases.

11 pages, 13 figures included. Final version as published

References in corpus (6)

• Anomalous transport in the crowded world of biological cells
• Asymptotic behavior of self-affine processes in semi-infinite domains
• Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries
• Reflected fractional Brownian motion in one and higher dimensions
• Probability distributions for polymer translocation
• Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls