The subordinated processes controlled by a family of subordinators and corresponding Fokker-Planck type equations

In this work, we consider subordinated processes controlled by a family of subordinators which consist of a power function of time variable and a negative power function of $\alpha-$stable random variable. The effect of parameters in the subordinators on the subordinated process is discussed. By suitable variable substitutions and Laplace transform technique, the corresponding fractional Fokker-Planck-type equations are derived. We also compute their mean square displacements in a free force field. By choosing suitable ranges of parameters, the resulting subordinated processes may be subdiffusive, normal diffusive or superdiffusive.