Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries

Kay Jörg Wiese ¹, Satya N. Majumdar ², Alberto Rosso ²

Physical Review E 83 (2011) 061141

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations ~ t^{2H} + s^{2H} – |t-s|^{2H}, where H, with 0 0 (near the absorbing boundary), while R(y) ~ y^gamma exp(-y^2/2) as y -> oo, with phi = 1 – 4 epsilon + O(epsilon^2) and gamma = 1 – 2 epsilon + O(epsilon^2). Our epsilon-expansion result confirms the scaling relation phi = (1-H)/H proposed in Ref. [28]. We verify our findings via numerical simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.

• 1. Laboratoire de Physique Théorique de l’ENS (LPTENS),
  CNRS : UMR8549 – Université Paris VI – Pierre et Marie Curie – Ecole Normale Supérieure de Paris – ENS Paris
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
  CNRS : UMR8626 – Université Paris XI – Paris Sud