  The standard diffusive spreading, characterized by a Gaussian distribution
with mean square displacement that grows linearly with time, can break down,
for instance, under the presence of correlations and heterogeneity. In this
work, we consider the spread of a population of fractional (long-time
correlated) Brownian walkers, with time-dependent and heterogeneous
diffusivity. We aim to obtain the possible scenarios related to these
individual-level features from the observation of the temporal evolution of the
population spatial distribution. We develop and discuss the possibility and
limitations of this connection for the broad class of self-similar diffusion
processes. Our results are presented in terms of a general framework, which is
then used to address well-known processes, such as Laplace diffusion, nonlinear
diffusion, and their extensions.
