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We consider a particle evolving in the quadratic potential and subject to atime-inhomogeneous frictional force and to a random force. The couple of itsvelocity and position is solution to a stochastic differential equation drivenby an $\alpha$-stable L{\'e}vy process with $\alpha \in (1,2]$ and thefrictional force is of the form $t^{-\beta}\text{sgn}(v)|v|^\gamma$. Weidentify three regimes for the behavior in long-time of the couplevelocity-position with a suitable rescaling, depending on the balance betweenthe frictional force and the index of stability $\alpha$ of the noise.

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