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 a symmetric $ \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 [ABSTRACT FROM AUTHOR]