Functionals of fractional Brownian motion and the three arcsine laws

TL;DR

This paper develops a perturbation expansion to analytically evaluate three key functionals of fractional Brownian motion, generalizing the three arcsine laws of standard Brownian motion and confirming results with numerical simulations.

Contribution

It introduces a perturbation method to derive analytical expressions for functionals of fractional Brownian motion, extending classical arcsine laws to non-Markovian processes.

Findings

Derived expressions for probabilities of three functionals as an epsilon expansion.

Found that the three probabilities differ for H ≠ 1/2 and coincide at H=1/2.

Validated analytical results with high-precision numerical simulations.

Abstract

Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent $H\in [0,1]$, generalising standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for non-equilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many non-trivial observables analytically: We generalize the celebrated three arcsine-laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in $\epsilon = H-\tfrac{1}{2}$, up to second order. We find that the three probabilities are different, except for $H=\tfrac{1}{2}$ where they coincide. Our results are confirmed to high precision by numerical simulations.