  In this paper we study first exit times from a bounded domain of a gradient
dynamical system $\dot Y_t=-\nabla U(Y_t)$ perturbed by a small multiplicative
L\'evy noise with heavy tails. A special attention is paid to the way the
multiplicative noise is introduced. In particular we determine the asymptotics
of the first exit time of solutions of It\^o, Stratonovich and Marcus canonical
SDEs.
