Reflected Fractional Brownian Motion on expanding one-dimensional intervals

Fractional Brownian motion (FBM) is a non-Markovian stochastic process characterized by long-range tem-

poral correlations. Its mean square displacement (MSD) scales as ⟨x²⟩ ∼ t^α with anomalous diffusion exponent

α. Previous studies of FBM confined by reflecting walls have shown deviation from the uniform distribution

expected for normal diffusion. The particle probability density exhibits a power law divergence near the wall

for superdiffusive FBM (α > 1), while for the subdiffusive (α < 1) case it is depleted near the wall [1, 2]. Here

we study FBM on one-dimensional intervals with reflecting boundaries that expand with time to investigate the

particle density near moving boundaries. We perform large-scale computer simulations to compute the MSD

and the probability density in the regime where the interval expands more slowly than the diffusing particle

cloud. We find that the probability density near the boundaries follows the same power-law behavior, P ∼ x^κ

as it does for fixed intervals.

Recently, FBM with reflecting boundaries has been employed to model the distribution of serotonergic fibers in

the brain as it captures the observed accumulation of fibers near brain surfaces in mice [3]. Our results suggest

that fiber accumulation near boundaries in a growing brain follows the same scaling behavior as in a static brain

model.