Tempered fractional Brownian motion on finite intervals

TL;DR

This paper studies how tempering fractional Brownian motion affects particle distribution near boundaries, revealing that exponential tempering creates a characteristic zone size while power-law tempering causes more complex boundary behaviors.

Contribution

It provides a detailed analysis of boundary effects in tempered fractional Brownian motion confined to finite intervals, highlighting differences between exponential and power-law tempering.

Findings

Exponential tempering introduces a characteristic size for accumulation zones.

Power-law tempering results in complex boundary behaviors differing between superdiffusive and subdiffusive cases.

Tempering does not alter the probability density's functional form near the wall.

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.