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Fractional Brownian motion is a non-Markovian Gaussian process indexed by theHurst exponent $H\in [0,1]$, generalising standard Brownian motion to accountfor anomalous diffusion. Functionals of this process are important forpractical applications as a standard reference point for non-equilibriumdynamics. We describe a perturbation expansion allowing us to evaluate manynon-trivial observables analytically: We generalize the celebrated threearcsine-laws of standard Brownian motion. The functionals are: (i) the fractionof time the process remains positive, (ii) the time when the process lastvisits the origin, and (iii) the time when it achieves its maximum (orminimum). We derive expressions for the probability of these three functionalsas an expansion in $\epsilon = H-\tfrac{1}{2}$, up to second order. We findthat the three probabilities are different, except for $H=\tfrac{1}{2}$ wherethey coincide. Our results are confirmed to high precision by numericalsimulations.

Functionals of fractional Brownian motion and the three arcsine laws