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Starting from a classical-mechanics stochastic model encoded in a Langevinequation, we derive the natural diffusion equation associated with threeclasses of multiscale spacetimes (with weighted, ordinary, and "q-Poincar\'e"symmetries). As a consistency check, the same result is obtained by inspectingthe propagation of a quantum-mechanical particle in a disordered environment.The solution of the diffusion equation displays a time-dependent diffusioncoefficient and represents a probabilistic process, classified according to thestatistics of the noise in the Langevin equation. We thus illustrate, also withpictorial aids, how spacetime geometries can be more completely catalogued notonly through their Hausdorff and spectral dimension, but also by a stochasticprocess. The spectral dimension of multifractional spacetimes is then computedand compared with what was found in previous studies, where a diffusionequation with some open issues was assumed rather than derived. These issuesare here discussed and solved, and they point towards the model withq-Poincar\'e symmetries.

Spectral dimension and diffusion in multi-scale spacetimes