Stochastic foundations of $g$-subdiffusion process

TL;DR

This paper establishes the stochastic foundations of the $g$-subdiffusion process by deriving its governing equation from a modified Continuous Time Random Walk model, clarifying its stochastic interpretation.

Contribution

It provides the first derivation of the $g$-subdiffusion equation from a stochastic model, linking it to a modified CTRW framework.

Findings

Derived the $g$-subdiffusion equation from a modified CTRW model.

Provided a stochastic interpretation of the $g$-subdiffusion process.

Enhanced understanding of diffusion processes with evolving characteristics.

Abstract

Recently, in the paper: T. Koszto{\l}owicz and A. Dutkiewicz, Phys. Rev. E \textbf{104}, 014118 (2021) the $g$--subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ has been considered. This equation offers new possibilities for modelling diffusion such as a process in which a type of diffusion evolves continuously over time. However, the equation has not been derived from a stochastic model and the stochastic interpretation of $g$--subdiffusion has been unknown. In this paper we show stochastic foundations of this process. We derive the equation by means of a modified Continuous Time Random Walk model. Interpretation of the $g$--subdiffusion process is also discussed.