In this talk, we prove a convergence theorem for singular perturbations problems for a class of fully nonlinear parabolic partial differential equations (PDEs) with ergodic structures. The limit function is represented as the viscosity solution to a fully nonlinear degenerate PDEs. Our approach is mainly based on G-ergodic theory. Note that the stochastic differential equations driven by G-Brownian motion (G-SDEs) have the unique invariant and ergodic expectations. However, the invariant expectations may not coincide with the ergodic expectations, which is different from the classical case. As a byproduct, we also establish the averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs) with two time-scales. The results extend Khasminskii’s averaging principle to nonlinear case.

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