  .Stochastic models based on random diffusivities, such as the
diffusing-diffusivity approach, are popular concepts for the description of
non-Gaussian diffusion in heterogeneous media. Studies of these models
typically focus on the moments and the displacement probability density
function. Here we develop the complementary power spectral description for a
broad class of random diffusivity processes. In our approach we cater for
typical single particle tracking data in which a small number of trajectories
with finite duration are garnered. Apart from the diffusing-diffusivity model
we study a range of previously unconsidered random diffusivity processes, for
which we obtain exact forms of the probability density function. These new
processes are different versions of jump processes as well as functionals of
Brownian motion. The resulting behaviour subtly depends on the specific model
details. Thus, the central part of the probability density function may be
Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential,
log-normal or even power-law forms. For all these models we derive analytically
the moment-generating function for the single-trajectory power spectral
density. We establish the generic $1/f^2$-scaling of the power spectral density
as function of frequency in all cases. Moreover, we establish the probability
density for the amplitudes of the random power spectral density of individual
trajectories. The latter functions reflect the very specific properties of the
different random diffusivity models considered here. Our exact results are in
excellent agreement with extensive numerical simulations.
