Asymptotic behaviours of multiscale multivalued stochastic systems with small noises

In this paper, we consider asymptotic behaviours of multiscale multivalued stochastic systems with small noises. First of all, for general, fully coupled systems for multivalued stochastic differential equations of slow and fast motions with small noises in the slow components, we prove an averaging principle in the strong convergence sense. Moreover, a convergence rate is given in a special case. Next, for these systems, we establish the large deviation principle by the weak convergence approach. Then, for a special case, the rate function is explicitly characterized. Finally, we explain our results with an example.