  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 L{\'e}vy 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.
