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

Abstract

We consider a dynamical system described by the differential equation Ẏt=−U'(Yt) with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity ɛ to obtain the stochastic differential equation dXtɛ=−U'(Xt−ɛ) dt+ɛ dLt. The process L is a symmetric Lévy process whose jump measure ν has exponentially light tails, ν([u, ∞))∼exp(−uα), 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 ɛ→0, the law of the first exit time σx, x∈(−1, 1), has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index α=1, namely, ln Eσ∼ɛ−α for 0<α<1, whereas ln Eσ∼ɛ−1|ln ɛ|1−1/α for α>1.