Convergence of martingale solutions to the hybrid slow-fast system

This paper is devoted to studying the weak convergence for a slow-fast system with jumps modulated by Markovian switching regimes with the martingale method. However, due to the co-existence of fast component and Markovian switching regimes, the martingale method and perturbed test functions cannot be applied directly. In this situation, a combination of perturbed test functions and the time discretization is applied efficiently. And the choice of appropriate perturbed test functions, which are related to the averaged coefficients, plays a decisive role. Our results also cover the case of slow-fast system without Markovian switching regimes. Finally, some examples are presented, and numerical simulations are carried out to observe a good agreement.