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

We consider a dynamical system described by the differential equation ˙ Y t = −U ′ (Y t) with a unique stable point at the origin. We perturb the system by Lévy noise of intensity ε, to obtain the stochastic differential equation dX ε t = −U ′ (X ε t−)dt + εdL t. The process L is a symmetric Lévy process whose jump measure ν has exponentially light tails, ν([u, ∞)) ∼ exp(−u α), α > 0, u → ∞. 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 ε → 0, the law of the first exit time σ x , x ∈ (−1, 1), is exponential with the mean value exhibiting an intriguing phase transition at the critical index α = 1, namely ln Eσ ∼ ε −α for 0 < α < 1, whereas ln Eσ ∼ ε −1 | ln ε| 1− 1 α for α > 1.