  We consider a description of the stochastic oscillations of the general
relativistic accretion disks around compact astrophysical objects interacting
with their external medium based on a generalized Langevin equation with
colored noise, which accounts for the general memory and retarded effects of
the frictional force, and on the fluctuation-dissipation theorem. The presence
of the memory effects influences the response of the disk to external random
interactions, and modifies the dynamical behavior of the disk, as well as the
energy dissipation processes. The generalized Langevin equation of the motion
of the disk in the vertical direction is studied numerically, and the vertical
displacements, velocities and luminosities of the stochastically perturbed
disks are explicitly obtained for both the Schwarzschild and the Kerr cases.
The Power Spectral Distribution (PSD) of the disk luminosity is also obtained.
As a possible astrophysical application of the formalism we investigate the
possibility that the Intra Day Variability (IDV) of the Active Galactic Nuclei
(AGN) may be due to the stochastic disk instabilities. The perturbations due to
colored/nontrivially correlated noise induce a complicated disk dynamics, which
could explain some astrophysical observational features related to disk
variability.
