  A considerable number of systems have recently been reported in which
Brownian yet non-Gaussian dynamics was observed. These are processes
characterised by a linear growth in time of the mean squared displacement, yet
the probability density function of the particle displacement is distinctly
non-Gaussian, and often of exponential (Laplace) shape. This apparently
ubiquitous behaviour observed in very different physical systems has been
interpreted as resulting from diffusion in inhomogeneous environments and
mathematically represented through a variable, stochastic diffusion
coefficient. Indeed different models describing a fluctuating diffusivity have
been studied. Here we present a new view of the stochastic basis describing
time dependent random diffusivities within a broad spectrum of distributions.
Concretely, our study is based on the very generic class of the generalised
Gamma distribution. Two models for the particle spreading in such random
diffusivity settings are studied. The first belongs to the class of generalised
grey Brownian motion while the second follows from the idea of diffusing
diffusivities. The two processes exhibit significant characteristics which
reproduce experimental results from different biological and physical systems.
We promote these two physical models for the description of stochastic particle
motion in complex environments.
