Deriving sub-diffusion equations

Benoît Perthame; Min Tang

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arXiv · 2507.20602

Deriving sub-diffusion equations

Benoît Perthame·Min Tang

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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.

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Benoît Perthame, Min Tang. 2025-07-28. Deriving sub-diffusion equations. https://arxiv.org/abs/2507.20602

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