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 the 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 $\varepsilon\to0$, the law of the first exit time $\sigma_x$, $x\in(-1,1)$, has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index $\alpha=1$, namely, $\ln\mathbf{ E}\sigma\sim\varepsilon^{ -\alpha}$ for $0<\alpha<1$, whereas $\ln\mathbf{ E}\sigma\sim\varepsilon^{ - 1}|\ln\varepsilon|^{ 1-{ 1}/{ \alpha}}$ for $\alpha>1$.
Categories: math.PR