Homogenization of semi-linear PDEs with discontinuous effective coefficients

TL;DR

This paper investigates the asymptotic behavior of semi-linear PDEs with discontinuous effective coefficients, using probabilistic methods and $L^p$-viscosity solutions to handle non-classical cases.

Contribution

It introduces a novel approach to analyze semi-linear PDEs with discontinuous coefficients without assuming periodicity or ergodicity, utilizing weak convergence and $L^p$-viscosity solutions.

Findings

Established existence of $L^p$-viscosity solutions for the averaged PDE.

Proved weak continuity of the limit diffusion process.

Linked the PDE limit to backward stochastic differential equations.

Abstract

We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in \`{C}esaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of "$L^p-$viscosity solution" introduced in \cite{CCKS}. We use BSDEs techniques to establish the existence of $L^p-$viscosity solution for the averaged PDE. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of $L^p$-viscosity solution.