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We study the exit problem of solutions of the stochastic differential equation $dX_t^varepsilon=-U'(X_t^\varepsilon),dt+ \varepsilon,dL_t$ from bounded or unbounded intervals which contain the unique asymptotically stable critical point of the deterministic dynamical system $\dot Y_t=-U'(Y_t)$. The process $L$ is composed of a standard Brownian motion and a symmetric $\alpha$-stable L'evy process. Using probabilistic estimates we show that in the small noise limit $\varepsilon \to 0$, the exit time of $X^\varepsilon$ from an interval is an exponentially distributed random variable and determine its expected value. Due to the heavy-tail nature of the $\alpha$-stable component of $L$, the results differ strongly from the well known case in which the deterministic dynamical system undergoes purely Gaussian perturbations (Kramers' law, Freidlin--Wentzel theory).

We also discuss the physical motivation for this problem which comes from theanalysis of Greenland paleoclimatic ice-core data.