Probabilistic approach to singular perturbations of viscosity solutions to nonlinear parabolic PDEs

Abstract

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 the one of Khasminskii [25] to nonlinear case.