Fractal spacetimes in stochastic gravity? - views from anomalous diffusion and the correlation hierarchy

We explore in stochastic gravity theory whether non-Gaussian noises from the higher order correlation functions of the stress tensor for quantum matter fields when backreacting on the spacetime may reveal hints of multi-scale structures. Anomalous diffusion may depict how a test particle experiences in a fractal spacetime. The hierarchy of correlations in quantum matter field induces the hierarchy of correlations in geometric objects via the set of Einstein-Langevin equations for each correlation order. This correlation hierarchy kinetic theory conceptual framework, aided by the characteristics of stochastic processes, may serve as a conduit for connecting the low energy Bottom-Up approach with the Top-Down theories of quantum gravity which predict the appearance of fractal spacetimes at the Planck scale.