Large deviations for Lévy diffusions in the small noise regime

Abstract

This paper 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 [Formula: see text] and with accelerated jumps by intensity [Formula: see text]. We establish Freidlin–Wentzell estimates for the slow process of the multiscale system in the small noise limit [Formula: see text] 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.