First exit times for L\'evy-driven diffusions with exponentially light jumps

TL;DR

This paper analyzes the first exit times of Le9vy-driven diffusions with exponentially light jumps, revealing a phase transition in the mean exit time behavior at a critical tail index.

Contribution

It provides a detailed asymptotic analysis of exit times for systems driven by symmetric Le9vy noise with exponential tails, highlighting a phase transition at =1.

Findings

Exponential tail behavior of the first exit time distribution.

Distinct asymptotic regimes for <1 and >1.

Identification of a phase transition at =1.

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