We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coecients have averages in the Cesaro sense. In such a case, the averaged coecients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic dierential equation (BSDE) in the Jakubowski $S$-topology to derive the averaged PDE. However, since the averaged coecients are discontinuous, the classical viscosity solution is not dened for the averaged PDE. We then use the notion of "$L_p$-viscosity solution" introduced in . The existence of $L_p$-viscosity solution to the averaged PDE is proved here by using BSDEs techniques.
"Homogenization of semilinear PDEs with discontinuous averaged coefficients." Electron. J. Probab. 14 477 - 499, 2009. https://doi.org/10.1214/EJP.v14-627