Averaging principle for two time-scale stochastic differential equations with fast component in noncompact space

The asymptotic behavior for fully coupled multiscale stochastic systems becomes much complicated when the fast processes do not locate in a compact space. An example is constructed to show that the averaged coefficients may become discontinuous even they are originally Lipschitz continuous when the fast process locate in a noncompact space. This work aims to reveal the impact of ergodicity of the fast process on the establishment of the averaging principle. The crucial point is to characterize the continuous dependence of the invariant probability measure on parameters related to the slow process with respect to various distances in the Wasserstein space.