arXiv:1908.03481v1 [math.DS] 9 Aug 2019 Large deviations for stochastic systems of slow-fast diffusions with non-Gaussian L´ evy noises ✩ Shenglan Yuan a , Ren´ e Schilling b , Jinqiao Duan c a Center for Mathematical Sciences in Huazhong University of Sciences and Technology, Wuhan 430074, China e-mail: shenglanyuan@hust.edu.cn b Institut f¨ ur Mathematische Stochastik in Technische Universit¨ at Dresden, Dresden, D-01069, Germany e-mail: rene.schilling@tu-dresden.de c Department of Applied Mathematics in Illinois Institute of Technology, Chicago, IL 60616, USA e-mail: duan@iit.edu 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 func- tionals 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. Keywords: Large deviations, slow-fast dynamical system, L´ evy noises, viscosity solutions, comparison principle. 2010 MSC: 60F10 (primary), 37H10 (secondary) 1. Introduction Many dynamical systems under random influences often involve the in- terplay of slow and fast variables. For instance, climate-weather interaction ✩ This work was partly supported by the NSFC grants 11701200, 11771161, 11771449 and 0118011074. Preprint submitted to August 12, 2019