First exit times of SDEs driven by stable Lévy processes

We study the exit problem of solutions of the stochastic differential equation dXtε=−U′(Xtε)dt+εdLt from bounded or unbounded intervals which contain the unique asymptotically stable critical point of the deterministic dynamical system Ẏt=−U′(Yt). The process LL is composed of a standard Brownian motion and a symmetric αα-stable Lévy process. Using probabilistic estimates we show that, in the small noise limit ε→0ε→0, the exit time of XεXε from an interval is an exponentially distributed random variable and determine its expected value. Due to the heavy-tail nature of the αα-stable component of LL, the results differ strongly from the well known case in which the deterministic dynamical system undergoes purely Gaussian perturbations.