Functionals of fractional Brownian motion and the three arcsine laws
Tridib Sadhu1, Kay Jörg Wiese2
1Tata Institute of Fundamental Research, Mumbai 400005, India.
2CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC,
24 rue Lhomond, 75005 Paris, France.
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.
arXiv:2103.09032 [pdf]
Phys. Rev. E 104 (2021) 054112 [pdf]
Copyright (C)
by Kay Wiese. Last edited March 17, 2021.