Fractional exponents

In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process BH(t) on [0, T], that starts at zero, has expectation zero for all t in [0, T], and has the following covariance function:




E
[

B

H


(
t
)

B

H


(
s
)
]
=



1
2



(

|

t


|


2
H


+

|

s


|


2
H


−

|

t
−
s


|


2
H


)
,


{\displaystyle E[B_{H}(t)B_{H}(s)]={\tfrac {1}{2}}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),}
where H is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Mandelbrot & van Ness (1968).
The value of H determines what kind of process the fBm is:

if H = 1/2 then the process is in fact a Brownian motion or Wiener process;
if H > 1/2 then the increments of the process are positively correlated;
if H < 1/2 then the increments of the process are negatively correlated.The increment process, X(t) = BH(t+1) − BH(t), is known as fractional Gaussian noise.
There is also a generalization of fractional Brownian motion: n-th order fractional Brownian motion, abbreviated as n-fBm. n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order n are stationary. For n = 1, n-fBm is classical fBm.
Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown; fractional Gaussian noise is named after mathematician Carl Friedrich Gauss.

View More On Wikipedia.org