Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
Kay Jörg Wiese1, Satya N. Majumdar2, Alberto Rosso1,2
1CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure,
24 rue Lhomond, 75005 Paris, France.
2CNRS-Laboratoire de Physique Théorique et Modèles Statistiques,
Université Paris-Sud, 91405 Orsay, France.
Abstract
Fractional Brownian motion is a Gaussian process x(t) with zero
mean and two-time correlations ⟨x(t1)x(t2)⟩ = D (t12H+t22H-|t1-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 ≠ 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-H R+(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 + ε, to
calculate the scaling function R+(y) to first order in ε. We find that R+(y) behaves as R+(y) ~ yφ as y → 0 (near the absorbing boundary), while
R+(y) ~ yγ exp(-y2/2) as y → ∞, with φ = 1 - 4 ε + O(ε2) and γ = 1 - 2 ε + O(ε2). Our ε-expansion
result confirms the scaling relation φ = (1-H)/H proposed in
Ref. [48]. We verify our findings via numerical
simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.
arXiv:1011.4807 [pdf]
Phys. Rev. E 83 (2011) 061141 [pdf]
Copyright (C) by Kay Wiese. Last edited November 22, 2010.