Stochastic foundations of $g$-subdiffusion process

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.