  We establish Freidlin-Wentzell results for a nonlinear ordinary differential
equation starting close to the stable state $0$, say, subject to a perturbation
by a stochastic integral which is driven by an $\varepsilon$-small and
$(1/\varepsilon)$-accelerated L\'evy process with exponentially light jumps.
For this purpose we derive a large deviations principle for the stochastically
perturbed system using the weak convergence approach developed by Budhiraja,
Dupuis, Maroulas and collaborators in recent years. In the sequel we solve the
associated asymptotic first escape problem from the bounded neighborhood of $0$
in the limit as $\varepsilon \rightarrow 0$ which is also known as the Kramers
problem in the literature.
