Fractional Brownian Motion in Confined Geometries

Fractional Brownian motion (FBM) is a Gaussian stochastic process with memory: the increments (steps) are long-range correlated or anticorrelated in time. When allowed to propagate in a confined geometry with reflecting walls, particles exhibiting FBM tend to accumulate at the domain’s boundaries (if the steps are positively correlated) or near the center of the domain (if the steps are negatively correlated). It has been conjectured that, in a confined geometry, the probability density at the boundary will vary as P(x) ~ x^κ, where x is the distance from the wall. We confirm this prediction by performing large-scale numerical simulations of reflected FBM in confined geometries of one, two, and three dimensions. In addition, we measure the exponent κ for each geometry and determine its dependence on the anomalous diffusion coefficient α [1].
[1] T. Vojta, S. Halladay, S. Skinner, S. Janusonis, T. Guggenberger, and R. Metzler, Phys. Rev. E 102, 032108 (2020).