  The emergence of the exit events from a bounded domain containing a stable
fixed point induced by non-Gaussian L\'evy fluctuations plays a pivotal role in
practical physical systems. In the limit of weak noise, we develop a
Hamiltonian formalism under the L\'evy fluctuations with exponentially light
jumps for one- and two-dimensional stochastic dynamical systems. This formalism
is based on a recently proved large deviation principle for dynamical systems
under non-Gaussian L\'evy perturbations. We demonstrate how to compute the most
probable exit path and the quasi-potential by several examples. Meanwhile, we
explore the impacts of the jump measure on the quasi-potential quantitatively
and on the most probable exit path qualitatively. Results show that the
quasi-potential can be well estimated by an approximate analytical expression.
Moreover, we discover that although the most probable exit paths are analogous
to the Gaussian case for the isotropic noise, the anisotropic noise leads to
significant changes of the structure of the exit paths. These findings shed
light on the underlying qualitative mechanism and quantitative feature of the
exit phenomenon induced by non-Gaussian noise.
