Riemann–Liouville and Weyl fractional oscillator processes

Two types of oscillator processes can be obtained as solutions to fractional Langevin equation based on Riemann–Liouville and Weyl fractional integro-differential operators. The relation between these fractional oscillator processes and the corresponding fractional Brownian motion is considered. Generalization of the Weyl fractional oscillator process to positive temperature can be carried out and its partition function can be calculated using the zeta function regularization method.