Large Deviations for L\'evy Diffusions in small regime

TL;DR

This paper develops large deviations estimates for a two-time scale Lévy-driven stochastic system with small noise, solving the Kramers problem and analyzing exit times using weak convergence and stochastic averaging techniques.

Contribution

It introduces a large deviations framework for Lévy diffusions in multiscale systems, applying weak convergence methods to establish Freidlin-Wentzell estimates and solve exit time problems.

Findings

Established Freidlin-Wentzell estimates for slow processes

Reduced large deviations to stochastic averaging principles

Solved exit time and exit locus problems in small noise limit

Abstract

This article concerns the large deviations regime and the consequent solution of the Kramers problem for a two-time scale stochastic system driven by a common jump noise signal perturbed in small intensity $\varepsilon>0$ and with accelerated jumps by intensity $\frac{1}{\varepsilon}$. We establish Freidlin-Wentzell estimates for the slow process of the multiscale system in the small noise limit $\varepsilon \rightarrow 0$ using the weak convergence approach to large deviations theory. The core of our proof is the reduction of the large deviations principle to the establishment of a stochastic averaging principle for auxiliary controlled processes. As consequence we solve the first exit time/ exit locus problem from a bounded domain containing the stable state of the averaged dynamics for the family of the slow processes in the small noise limit.