The longest excursion of fractional Brownian motion : numerical evidence of non-Markovian effects

Reinaldo Garcia-Garcia 1, Alberto Rosso 2, Gregory Schehr 3

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 81 (2010) 010102(R)

We study, using exact numerical simulations, the statistics of the longest excursion l_{\max}(t) up to time t for the fractional Brownian motion with Hurst exponent 0 \propto Q_\infty t where Q_\infty \equiv Q_\infty(H) depends continuously on H, and in a non trivial way. These results are compared with exact analytical results obtained recently for a renewal process with an associated persistence exponent \theta = 1-H. This comparison shows that Q_\infty(H) carries the clear signature of non-Markovian effects for H\neq 1/2. The pre-asymptotic behavior of < l_{\max}(t)> is also discussed.

  • 1. Centro Atomico de Bariloche,
    Centro Atomico de Bariloche
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3. Laboratoire de Physique Théorique d'Orsay (LPT),
    CNRS : UMR8627 – Université Paris XI - Paris Sud