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## Fractional Brownian motion in a finite interval

• Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising resultsFractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

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Author: Tobias Guggenberger, Gianni PagniniORCiD, Thomas Vojta, Ralf MetzlerORCiDGND urn:nbn:de:kobv:517-opus4-436665 https://doi.org/10.25932/publishup-43666 1866-8372 Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe correlations effect depletion or accretion zones of particles near boundaries Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe (755) Postprint English 2019/10/17 2019 Universität Potsdam 2019/10/17 anomalous diffusion; fractional Brownian motion; reflecting boundary conditions 755 13 New Journal of Physics 21 (2019) Art. 022002 DOI: 10.1088/1367-2630/ab075f Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie 5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik Referiert Open Access Creative Commons - Namensnennung, 3.0 Deutschland Bibliographieeintrag der Originalveröffentlichung/Quelle