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The asymptotic behavior for fully coupled multiscale stochastic systemsbecomes much complicated when the fast processes do not locate in a compactspace. An illustrative example is constructed, which shows that the averagedcoefficients may become discontinuous even they are originally Lipschitzcontinuous. We aim to reveal the impact of the ergodicity of the fast processon the limit process and the averaging principle. The crucial point is thecontinuous dependence on parameters related to the slow process for theinvariant probability measures in various distances over the Wasserstein space.

Continuous dependence on parameters for invariant measures and its application to multiscale stochastic systems