Langevin picture of the g-diffusion process and its ergodic property

In the recent literature, the g-diffusion equation involving Caputo fractional derivative with respect to another function has been proposed to describe subdiffusion in a medium having an evolving structure. In this case, a continuous transition between different subdiffusive regimes may occur. To study the underlying process of the g-diffusion equation, we propose a Langevin equation coupled with a subordinator involving two time-changing mechanisms to describe the microscopic motion of the g-diffusion process, on which we derive the same Fokker-Planck equation containing the Caputo fractional derivative with respect to another function. We further develop the theories of evaluating the common quantities in the framework of a Langevin equation, such as ensemble-averaged mean-squared displacement, time-averaged mean-squared displacement (TAMSD), scatter of TAMSD, and ergodicity breaking parameter. The proposed Langevin equation and the new subordinator provide a new perspective for studying the g-diffusion process and further improve the theory of the anomalous diffusion phenomena in a complex nonstatic environment.