Fractional Brownian motion with a reflecting wall

Abstract

Fractional Brownian motion, a stochastic process with long-time correlations between its increments, is a prototypical model for anomalous diffusion. We analyze fractional Brownian motion in the presence of a reflecting wall by means of Monte Carlo simulations. While the mean-square displacement of the particle shows the expected anomalous diffusion behavior $⟨x^2⟩\sim t^{\alpha }$, the interplay between the geometric confinement and the long-time memory leads to a highly non-Gaussian probability density function with a power-law singularity at the barrier. In the superdiffusive case, $\alpha >1$, the particles accumulate at the barrier leading to a divergence of the probability density. For subdiffusion, $\alpha <1$, in contrast, the probability density is depleted close to the barrier. We discuss implications of these findings, in particular for applications that are dominated by rare events.

6 pages, 6 figures. Final version as published

References in corpus (8)

• Anomalous transport in the crowded world of biological cells
• Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
• "Diffusing diffusivity": A model for anomalous and "anomalous yet Brownian" diffusion
• Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
• Anomalous Polymer Dynamics Is Non-Markovian: Memory Effects and The Generalized Langevin Equation Formulation
• Persistent memory for a Brownian walker in a random array of obstacles
• Test of the diffusing-diffusivity mechanism using near-wall colloidal dynamics
• Enhanced rare region effects in the contact process with long-range correlated disorder

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