In this paper, 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 $G$-stochastic analysis argument. 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.

中文翻译：

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非线性抛物线偏微分方程粘度解奇异扰动的概率方法

在本文中，我们证明了一类具有遍历结构的全非线性抛物线偏微分方程 (PDE) 奇异摄动问题的收敛定理。极限函数表示为完全非线性退化偏微分方程的粘度解。我们的方法主要基于$G$-随机分析论证。作为副产品，我们还建立了随机微分方程的平均原理，由$G$-布朗运动（$G$-SDEs) 具有两个时间尺度。结果将 Khasminskii 的平均原理扩展到非线性情况。