  Many statistical properties of X-ray aperiodic variability from accreting
compact objects can be explained by the propagating fluctuations model applied
to the accretion disc. The mass accretion rate fluctuations originate from
variability of viscosity, which arises at every radius and causes local
fluctuations of the density. The fluctuations diffuse through the disc and
result in local variability of the mass accretion rate, which modulates the
X-ray flux from the inner disc in the case of black holes, or from the surface
in the case of neutron stars. A key role in the theoretical explanation of fast
variability belongs to the description of the diffusion process. The
propagation and evolution of the fluctuations is described by the diffusion
equation, which can be solved by the method of Green functions. We implement
Green functions in order to accurately describe the propagation of fluctuations
in the disc. For the first time we consider both forward and backward
propagation. We show that (i) viscous diffusion efficiently suppress
variability at time scales shorter than the viscous time, (ii) local
fluctuations of viscosity affect the mass accretion rate variability both in
the inner and the outer parts of accretion disc, (iii) propagating fluctuations
give rise not only to hard time lags as previously shown, but also produce soft
lags at high frequency similar to those routinely attributed to reprocessing,
(iv) deviation from the linear rms-flux relation is predicted for the case of
very large initial perturbations. Our model naturally predicts bumpy power
spectra.
