2021/03/16 by Tridib Sadhu, Kay Jörg Wiese · 14 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Brownian excursion #Brownian motion #Computer science #Diffusion process #Fractional Brownian motion #Fractional Differential Equations Solutions #Geometric Brownian motion #Hurst exponent #Mathematical analysis #Mathematical physics #Mathematics #Physics #Statistical Mechanics and Entropy #Statistical physics #Statistics #cond-mat.stat-mech #math-ph #math.MP #physics.data-an
paper · pdf · doi:10.1103/physreve.104.054112
published in Physical review. E 104(5), 054112 (American Physical Society) · 50 pages with 28 figures. For a supplemental Mathematica notebook (Ref[76]) see https://www.dropbox.com/s/jcgcjs5u2kfh4fm/SupplementalNotebook.nb?dl=0
arxiv created 2021/03/16 · openalex publication_date 2021/11/15 · arxiv updated 2021/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent H∈(0,1), generalizing standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for nonequilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many nontrivial 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 ɛ=H-1/2, up to second order. We find that the three probabilities are different, except for H=1/2, where they coincide. Our results are confirmed to high precision by numerical simulations.