Fractional Brownian motion and characterization of on-off
intermittency
Mingzhou Ding
Center for Complex Systems and Department of Mathematics,
Florida Atlantic University, Boca Raton, Florida 33431
Herein the term fractional Brownian motion is used to refer
to a class of long-range correlated random walks for which the mean
square displacement at large time t is proportional to 
with 0;SPMlt;H;SPMlt;1. 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 
. Theoretical arguments based on the fractal properties of
random walk trajectories are used to derive the result. We also
present supporting numerical simulations. Additional issues explored
include the distribution of the first passage time, modifications 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 dynamical mechanism for bursting.