Lack of anomalous diffusion in linear translationally-invariant systems determined by only one initial condition

TL;DR

This paper proves that in linear translationally-invariant systems, moments of the diffusion process evolve as polynomials of degree at most equal to their order, indicating no anomalous diffusion occurs under these conditions.

Contribution

It establishes a theoretical result linking translational invariance to the linear time evolution of moments in diffusion equations, ruling out anomalous diffusion.

Findings

Moments of degree α are polynomials of degree at most α.

Connected moments are at most linear functions of time.

Variance grows at most linearly with time.

Abstract

It is shown that as far as the linear diffusion equation meets both time- and space- translational invariance, the time dependence of a moment of degree $\alpha$ is a polynomial of degree at most equal to $\alpha$, while all connected moments are at most linear functions of time. As a special case, the variance is an at most linear function of time.