Subdiffusive fractional limit of a jump-renewal equation

TL;DR

This paper rigorously derives the subdiffusive, time-fractional diffusion limit from an age-structured jump-renewal model for continuous-time random walks with infinite mean waiting times. By a time-space rescaling and a careful measure-valued framework, the authors prove that the rescaled density $\rho_\varepsilon$ converges (along a subsequence) to a limit $\rho$ solving the time-fractional heat equation $\partial_t^\alpha \rho = D_\alpha\,\Delta\rho$, with diffusion coefficient $D_\alpha=\frac{\sigma^2}{2d\,\Psi\,\Gamma(1-\alpha)}$; this captures the subdiffusive MSD growth $\mathrm{MSD}(t)\sim t^\alpha$. The analysis relies on a robust renewal-structure toolkit, including convolution identities with $Y_\nu$ kernels and a mild-solution formulation for measure-valued densities, avoiding Laplace-transform injectivity assumptions. The results connect mesoscopic age-structured dynamics to a classical fractional PDE, enabling rigorous treatment of subdiffusion and potential inclusion of reaction terms in future work.

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.

Theorems & Definitions (27)

• Proposition 1
• Definition 2
• Theorem 3
• Proposition 4
• Lemma 5
• proof
• Corollary 6
• proof
• Lemma 7
• proof