# Large deviations for stochastic nonlinear systems of slow-fast diffusions with non-Gaussian L\'evy noises

## TL;DR

This paper proves a large deviation principle for slow variables in stochastic slow-fast systems driven by both Brownian and Lévy noises, analyzing their asymptotic behavior via viscosity solutions of integro-differential equations.

## Contribution

It establishes the large deviation principle for systems with non-Gaussian Lévy noises, extending the theory to include jump processes and complex noise interactions.

## Key findings

- Large deviation principle proven for slow variables
- Viscosity solutions used to analyze asymptotics
- Comparison principle verified for integro-differential equations

## Abstract

We establish the large deviation principle for the slow variables in slow-fast dynamical system driven by both Brownian noises and L\'evy noises. The fast variables evolve at much faster time scale than the slow variables, but they are fully inter-dependent. We study the asymptotics of the logarithmic functionals of the slow variables in the three regimes based on viscosity solutions to the Cauchy problem for a sequence of partial integro-differential equations. We also verify the comparison principle for the related Cauchy problem to show the existence and uniqueness of the limit for viscosity solutions.