  In this paper, we prove a convergence theorem for singular perturbations
problems for a class of fully nonlinear parabolic partial differential
equations 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-stochastic analysis argument. As a byproduct, we also establish the
averaging principle for stochastic differential equations driven by G-Brownian
motion with two time-scales. The results extend Khasminskii's averaging
principle to nonlinear case.
