Fokker–Planck type equations associated with fractional Brownian motion controlled by infinitely divisible processes

•The subordinated fractional Brownian motion with general inverse general subordinator is examined.•We present the corresponding fractional Fokker–Planck type equation.•We analyze main statistical properties of the considered system.•In a special case we classify the examined process as accelerating-subdiffusion system.•We present the simulation procedure.

In this paper we study the anomalous diffusion process driven by fractional Brownian motion delayed by general infinitely divisible subordinator. We show the analyzed process is the stochastic representation of the Fokker–Planck type equation that describes the probability density function of an introduced model. Moreover, we study main characteristics of the examined process, the first two moments, that allow us in special cases for classification of it as a system with accelerating-subdiffusion property.