Subdiffusive fractional limit of a jump-renewal equation

TL;DR

This paper studies a jump-renewal model with infinite mean waiting times, demonstrating that under rescaling, it converges to a time fractional subdiffusion equation, linking microscopic jump dynamics to macroscopic subdiffusive behavior.

Contribution

It establishes a rigorous connection between age-structured jump models and fractional subdiffusion equations through a rescaling limit.

Findings

The jump-renewal model converges to a fractional subdiffusion equation.

Infinite mean waiting times lead to subdiffusive scaling behavior.

Rescaling reveals the macroscopic limit of the microscopic process.

Abstract

In this paper, we consider an age-structured jump model that arises as a description of continuous time random walks with infinite mean waiting time between jumps. We prove that under a suitable rescaling, this equation converges in the long time large scale limit to a time fractional subdiffusion equation.