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

Abstract

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 Lévy 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 Lévy process whose jump measure $ν$ has exponentially light tails, $ν([u,\infty))\sim\exp(-u^α)$, $α>0$, $u\to \infty$. 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 $σ_x$, $x\in(-1,1)$, has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index $α=1$, namely, $\ln\mathbf{E}σ\sim\varepsilon^{-α}$ for $0<α<1$, whereas $\ln\mathbf{E}σ\sim\varepsilon^{- 1}|\ln\varepsilon|^{1-{1}/α}$ for $α>1$.