How rich is the class of multifractional Brownian motions?

The multifractional Brownian motion (MBM) processes are locally self-similar Gaussian processes. They extend the classical fractional Brownian motion processes BH={BH(t)}t∈RBH={BH(t)}t∈R by allowing their self-similarity parameter H∈(0,1)H∈(0,1) to depend on time.Two types of MBM processes were introduced independently by Peltier and Lévy-Vehel [Multifractional Brownian motion: definition and preliminary results, Technical Report 2645, Institut National de Recherche en Informatique et an Automatique, INRIA, Le Chesnay, France, 1995] and Benassi, Jaffard, Roux [Elliptic Gaussian random processes, Rev. Mat. Iber. 13(1) (1997) 19–90] by using time-domain and frequency-domain integral representations of the fractional Brownian motion, respectively. Their correspondence was studied by Cohen [From self-similarity to local self-similarity: the estimation problem, in: M. Dekking, J.L. Véhel, E. Lutton, C. Tricot (Eds.), Fractals: Theory and Applications in Engineering, Springer, Berlin, 1999]. Contrary to what has been stated in the literature, we show that these two types of processes have different   correlation structures when the function H(t)H(t) is non-constant.We focus on a class of MBM processes parameterized by (a+,a-)∈R2(a+,a-)∈R2, which contains the previously introduced two types of processes as special cases. We establish the connection between their time- and frequency-domain integral representations and obtain explicit expressions for their covariances. We show, that there are non-constant functions H(t)H(t) for which the correlation structure of the MBM processes depends non-trivially on the value of (a+,a-)(a+,a-) and hence, even for a given function H(t)H(t), there are an infinite number of MBM processes with essentially different distributions.