First exit times for Lévy-driven diffusions with exponentially light jumps.

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 $\alpha = 1$.