In this paper we consider the first exit problem of an overdamped
Lévy driven particle in a confining potential. We survey results
obtained in recent years from our work on the Kramers' times for
dynamical systems of this type with Lévy perturbations containing
heavy, and exponentially light jumps, and compare them to the well
known case of dynamical systems with Gaussian perturbations. It
turns out that exits induced by Lévy processes with jumps are
always essentially faster than Gaussian exits.
We consider a dynamical system described by the differential equation $\dot{Y}_t = -U^'(Y_t)$
with a unique stable point at the origin. We perturb the system by L\'evy noise of
intensity $\varepsilon$, to obtain the stochastic differential equation $dX^\varepsilon_t = -U^'(X^\varepsilon_{t-})dt + \varepsilon dL_t}.
The process $L$ is a symmetric L\'evy process whose jump measure $\nu$ has exponentially
light tails, $\nu([u, \infty))\sim exp(-u^\alpha), \alpha > 0, u \to\infty$. We study the first exit problem for
the trajectories of the solutions of the stochastic differential equation from the interval
$[-1, 1]$. In the small noise limit $\varepsilon\to 0$ we determine the law and the mean value of the
first exit time, to discover an intriguing phase transition at the critical index $\alpha = 1$.