2010/11/30 by Kay Jörg Wiese, Satya N. Majumdar, Alberto Rosso · 2 citations
Mathematics · Physics and Astronomy · #Boundary (topology) #Brownian motion #Combinatorics #Exponent #Fractional Brownian motion #Fractional Differential Equations Solutions #Geometry #Hurst exponent #Mathematical analysis #Mathematical physics #Mathematics #Order (exchange) #Physics #Quantum mechanics #Scaling #Scaling limit #Statistics #Stochastic processes and statistical mechanics #cond-mat.stat-mech #stochastic dynamics and bifurcation
paper · pdf · doi:10.1103/physreve.83.061141
published as Phys. Rev. E 83 (2011) 061141 · 16 pages, 8 figures; revised version 2 adds discussion on spatial small-distance cutoff
arxiv created 2011/04/11 · openalex publication_date 2011/06/24 · arxiv updated 2013/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations \ensuremath⟨x(t1)x(t2)\ensuremath⟩=D(t12H+t22H\ensuremath-|t1\ensuremath-t2|2H), where H, with 0<H<1, is called the Hurst exponent. For H=1/2, x(t) is a Brownian motion, while for H\ensuremath≠1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P+(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P+(x,t)~t^\ensuremath-HR+(x/tH). Our objective is to compute the scaling function R+(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H=1/2+\ensuremathε, to calculate the scaling function R+(y) to first order in \ensuremathε. We find that R+(y) behaves as R+(y)~y^\ensuremathφ as y\ensuremath→0 (near the absorbing boundary), while R+(y)~y^\ensuremathγexp(\ensuremath-y2/2) as y\ensuremath→\ensuremath∞, with \ensuremathφ=1\ensuremath-4\ensuremathε+O(\ensuremathε2) and \ensuremathγ=1\ensuremath-2\ensuremathε+O(\ensuremathε2). Our \ensuremathε-expansion result confirms the scaling relation \ensuremathφ=(1\ensuremath-H)/H proposed in Zoia, Rosso, and Majumdar [Phys. Rev. Lett. 102, 120602 (2009)]. We verify our findings via numerical simulations for H=2/3. The tools developed here are versatile, powerful, and adaptable to different situations.