It is an important physical property for a diffusion process to respond to an external force or a spatial confinement in experiments. This paper investigates the random diffusivity processes coupled with an inverse $\alpha$-stable subordinator which characterizes the trapping events in crowded environment, and studies its dynamics under the harmonic potential by evaluating the ensemble-averaged and time-averaged mean-squared displacements (MSDs). We build three kinds of coupled Langevin equations based on different physical realities of the external force in crowded environment, e.g., the harmonic potential only modifies the particle's motion at the moments of particle-making jumps, and that exerts an effect throughout the whole physical time, and the case with the validity of fluctuation-dissipation theorem. By comparing these three models, we find that the crowding trapping effect and temporal dynamic fluctuation play different roles in the MSDs. Nevertheless, we also observe an interesting phenomenon that the ratio between two MSDs exhibits the same scaling form, which perhaps reveals an underlying mechanism of the ergodicity breaking within the subordinated diffusion processes.

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• Anomalous diffusion
• Langevin equation