Strong and weak approximation for a two-time-scale stochastic quasi-geostrophic flow equation driven by Lévy processes

This work investigates the strong and weak approximation for a stochastic quasi-geostrophic flow equation with two time scales, where the slow component is coupled with a fast oscillation governed by a stochastic reaction-diffusion equation, and both are driven by Lévy noises. Employing Khasminskii's time discretization, we first prove that the slow component of the slow-fast system converges to the solution of the averaged equation in a strong sense with the help of an auxiliary process in small subintervals. Based on an asymptotic expansion of solutions for the Kolmogorov equation associated with the slow-fast system through a discontinuous path, we then decompose the weak solution with respect to the small parameter. By means of the components being determined recursively, we further establish the weak convergence from the original to the averaged dynamics.