  Many approaches to quantum gravity have resorted to diffusion processes to
characterize the spectral properties of the resulting quantum spacetimes. We
critically discuss these quantum-improved diffusion equations and point out
that a crucial property, namely positivity of their solutions, is not preserved
automatically. We then construct a novel set of diffusion equations with
positive semi-definite probability densities, applicable to Asymptotically Safe
gravity, Horava-Lifshitz gravity and Loop Quantum Gravity. These recover all
previous results on the spectral dimension and shed further light on the
structure of the quantum spacetimes by assessing the underlying stochastic
processes. Pointing out that manifestly different diffusion processes lead to
the same spectral dimension, we propose the probability distribution of the
diffusion process as a refined probe of quantum spacetime.
