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

TL;DR

This paper develops a probabilistic method to analyze singular perturbations in fully nonlinear parabolic PDEs, proving convergence to viscosity solutions and extending averaging principles to G-Brownian motion driven systems.

Contribution

It introduces a G-stochastic analysis approach for singular perturbations and extends Khasminskii's averaging principle to nonlinear stochastic differential equations.

Findings

Proved convergence of solutions to viscosity solutions in singular perturbation problems.

Established an averaging principle for G-Brownian motion driven SDEs.

Extended existing averaging principles to nonlinear and G-framework contexts.

Abstract

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.