  Starting from a classical-mechanics stochastic model encoded in a Langevin
equation, we derive the natural diffusion equation associated with three
classes of multiscale spacetimes (with weighted, ordinary, and "q-Poincar\'e"
symmetries). As a consistency check, the same result is obtained by inspecting
the propagation of a quantum-mechanical particle in a disordered environment.
The solution of the diffusion equation displays a time-dependent diffusion
coefficient and represents a probabilistic process, classified according to the
statistics of the noise in the Langevin equation. We thus illustrate, also with
pictorial aids, how spacetime geometries can be more completely catalogued not
only through their Hausdorff and spectral dimension, but also by a stochastic
process. The spectral dimension of multifractional spacetimes is then computed
and compared with what was found in previous studies, where a diffusion
equation with some open issues was assumed rather than derived. These issues
are here discussed and solved, and they point towards the model with
q-Poincar\'e symmetries.
