Fractional Brownian motion (FBM) is the unique self-similar Gaussian process with stationary increments. Through Lamperti’s transformation, it corresponds to the stationary Gaussian process , which is called the Ornstein-Uhlenbeck process associated to FBM. Besides , the authors also consider other two stationary Gaussian processes and , where has its covariance function given by
and is specified by a fractional Langevin equation which can be represented as
where is a normalization constant and is the white noise. Since the covariance function of is related to the modified Bessel function, the authors call the -Bessel process.
The authors compare the asymptotic behavior of the covariance functions and the spectral densities of , and and show that they have many common properties such as the covariance functions have similar local structures and their spectral density functions have similar asymptotic properties at large frequency. They argue that these stationary Gaussian processes can be regarded as the local stationary representations of FBM. They also consider the self-similar Gaussian processes and obtained from and via the (inverse) Lamperti transformation. They show that, even though and do not have stationary increments, the variances of their increments behave locally like that of FBM and the problem of long-range dependence can be studied. They give simulation of the sample paths of these Gaussian processes based on numerical Karhunen-Loève expansion.