# The Kramers problem for SDEs driven by small, accelerated L\'evy noise with exponentially light jumps

## TL;DR

This paper analyzes the escape behavior of nonlinear ODEs perturbed by small, accelerated Le9vy noise with exponentially light jumps, establishing large deviations principles and solving the Kramers problem.

## Contribution

It derives a large deviations principle for systems driven by accelerated Le9vy noise and solves the asymptotic escape problem, extending classical results to this new noise regime.

## Key findings

- Established Freidlin-Wentzell type results for the system.
- Derived a large deviations principle using the weak convergence approach.
- Solved the Kramers escape problem in the small noise limit.

## Abstract

We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state $0$, say, subject to a perturbation by a stochastic integral which is driven by an $\varepsilon$-small and $(1/\varepsilon)$-accelerated L\'evy process with exponentially light jumps. For this purpose we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel we solve the associated asymptotic first escape problem from the bounded neighborhood of $0$ in the limit as $\varepsilon \rightarrow 0$ which is also known as the Kramers problem in the literature.