Scaling limit of a kinetic inhomogeneous stochastic system in the quadratic potential

We consider a particle evolving in the quadratic potential and subject to a time-inhomogeneous frictional force and to a random force. The couple of its velocity and position is solution to a stochastic differential equation driven by an $\alpha$-stable Lévy process with $\alpha \in (1,2]$ and the frictional force is of the form $t^{-\beta}\text{sgn}(v)|v|^\gamma$. We identify three regimes for the behavior in long-time of the couple velocity-position with a suitable rescaling, depending on the balance between the frictional force and the index of stability $\alpha$ of the noise