  There is much confusion in the literature over Hurst exponent (H). The
purpose of this paper is to illustrate the difference between fractional
Brownian motion (fBm) on the one hand and Gaussian Markov processes where H is
different to 1/2 on the other. The difference lies in the increments, which are
stationary and correlated in one case and nonstationary and uncorrelated in the
other. The two- and one-point densities of fBm are constructed explicitly. The
two-point density does not scale. The one-point density for a semi-infinite
time interval is identical to that for a scaling Gaussian Markov process with H
different to 1/2 over a finite time interval. We conclude that both Hurst
exponents and one-point densities are inadequate for deducing the underlying
dynamics from empirical data. We apply these conclusions in the end to make a
focused statement about nonlinear diffusion.
