Distribution of the first return time in fractional Brownian motion and its application to the study of on-off intermittency

TLDR: The purpose of this paper is to show that the probability distribution of T scales with T as P(T)\ensuremath{\sim}$ T as H=1/2 and the application of the result to the characterization of on-off intermittency, a recently proposed mechanism for bursting.

Abstract: Herein, the term fractional Brownian motion is used to refer to a class of random walks with long-range correlated steps where the mean square displacement of the walker at large time t is proportional to ${\mathit{t}}^{2\mathit{H}}$ with 0H1. For ordinary Brownian motion we obtain H=1/2. Let T denote the time at which the random walker starting at the origin first returns to the origin. The purpose of this paper is to show that the probability distribution of T scales with T as P(T)\ensuremath{\sim}${\mathit{T}}^{\mathit{H}\mathrm{\ensuremath{-}}2}$. Theoretical arguments and numerical simulations are presented to support the result. Additional issues explored include modification to the power law distribution when the random walk is biased and the application of the result to the characterization of on-off intermittency, a recently proposed mechanism for bursting.