Deriving sub-diffusion equations

TL;DR

This paper rigorously derives sub-diffusion equations from microscopic models with age-structured waiting times, addressing mathematical challenges posed by long waiting times and non-integrable age equilibria.

Contribution

It provides a novel microscopic derivation of sub-diffusion equations using age-structured models and Laplace transforms, filling a gap in the mathematical understanding.

Findings

Derived sub-diffusion equations from microscopic models.

Addressed mathematical challenges due to non-integrable age equilibria.

Established a rigorous link between microscopic mechanisms and macroscopic sub-diffusion.

Abstract

Sub-diffusion equations are used in a large range of applications including fluids, plasma physics and biology. Their mathematical analysis is advanced even if a much larger literature addresses super-diffusions. The goal of this paper is to provide the microscopic mechanism and rigorous derivation of sub-diffusions when the waiting time distribution of particles follows an age-structured equation and jumps occur at each renewal. The major difficulty to recover sub-diffusions, unlike normal diffusions, is that the assumption of long waiting time implies lack of integrability for the age equilibrium. This prevents to establish strong a priori estimates. Here, the Laplace transform plays the role that Fourier transform plays for the more traditional case of fast diffusions.