The $\alpha$-stable time-changed fractional Ornstein-Uhlenbeck process and its Fokker-Planck equation

We consider the fractional Ornstein-Uhlenbeck process, solution of a stochastic differential equation driven by the fractional Brownian motion, and we study its time-changed version, obtained via an inverse $\alpha$-stable subordinator. We focus on the convergence of the probability density function as the Hurst index $H \to \frac{1}{2}$. The generalized fractional Fokker-Planck equation for such process is introduced and the class of subordinated solutions of such equation is studied, providing some uniqueness and isolation results and studying the convergence as $H \to \frac{1}{2}$.