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Dynamical systems driven by a general L\'evy stable noise are considered. Theinertia is included and the noise, represented by a generalisedOrnstein-Uhlenbeck process, has a finite relaxation time. A general linearproblem (the additive noise) is solved: the resulting distribution convergeswith time to the distribution for the white-noise, massless case. Moreover, amultiplicative noise is discussed. It can make the distribution steeper and thevariance, which is finite, depends sublinearly on time (subdiffusion). For asmall mass, a white-noise limit corresponds to the Stratonovich interpretation.On the other hand, the distribution tails agree with the Ito interpretation ifthe inertia is very large. An escape time from the potential well iscalculated.

Anomalous diffusion in systems driven by the stable Levy noise with a finite noise relaxation time and inertia