Integral representations and properties of operator fractional

Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar, and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time reversibility, the defining properties (i)-(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here through their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and by taking the Fourier transform of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of OFBM has a rigid structure, called here Dichotomy Principle. The notion of operator Brownian motions is also explored.