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This work studies the averaging principle for a fully coupled two time-scalesystem, whose slow process is a diffusion process and fast process is a purelyjumping process on an infinitely countable state space. The ergodicity of thefast process has important impact on the limit system and the averagingprinciple. We showed that under strongly ergodic condition, the limit systemadmits a unique solution, and the slow process converges in the L1-norm to thelimit system. However, under certain weaker ergodicity condition, the limitsystem admits a solution, but not necessarily unique, and the slow process canbe proved to converge weakly to a solution of the limit system.

Averaging principle for two time-scale regime-switching processes