Diffusion in multiscale spacetimes

TL;DR

This paper investigates diffusion processes in quantum geometries modeled as multiscale spacetimes, presenting new solutions and stochastic interpretations relevant to quantum spacetime structure.

Contribution

It introduces novel diffusion equations and solutions tailored for quantum spacetimes, extending probability and percolation theory to quantum geometric contexts.

Findings

Different types of diffusion equations are formulated and solved.

Stochastic processes are identified for quantum spacetime models.

Spectral-dimension profiles for multifractional spaces are constructed.

Abstract

We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples and the most general spectral-dimension profile of multifractional spaces is constructed.