We study the asymptotic behavior of the solution of semi-linear PDEs. Neither periodicity nor ergodicity assumptions are assumed. The coefficients admit only a limit in aCesaro sense. In such a case, the limit coefficients may have discontinuity. We use probabilistic approach based on weak convergence techniques for the associated backward stochastic differential equation in the S-topology. 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.

We prove an existence and uniqueness result for backward stochastic differential equations whose coefficients satisfy a stochastic monotonicity condition. In this setting, we deal with both constant and random terminal times. In the random case, the terminal time is allowed to take infinite values. But in a Markovian framework, that is coupled with a forward SDE, our result provides a probabilistic interpretation of solutions to nonlinear PDEs

International audienceWe establish an averaging principle for a family of solutions (Xε,Yε) := (X1,ε, X2,ε, Yε) of asystem of decoupled forward backward stochastic differential equations (SDE-BSDE for short) with a nullrecurrent fast component X1,ε. In contrast to the classical periodic case, we can not rely on an invariantprobability and the slow forward component X2,ε cannot be approximated by a diffusion process. Onthe other hand, we assume that the coefficients admit a limit in a Cesa`ro sense. In such a case, the limitcoefficients may have discontinuity. We show that the triplet (X1,ε, X2,ε, Yε) converges in law to thesolution (X1, X2,Y) of a system of SDE–BSDE, where X := (X1, X2) is a Markov diffusion which isthe unique (in law) weak solution of the averaged forward component and Y is the unique solution to theaveraged backward component. This is done with a backward component whose generator depends on thevariable z. As application, we establish an homogenization result for semilinear PDEs when the coefficientscan be neither periodic nor ergodic. We show that the averaged BDSE is related to the averaged PDE via aprobabilistic representation of the (unique) Sobolev W 1,2(R+ × Rd )–solution of the limit PDEs. Our d +1,locapproach combines PDE methods and probabilistic arguments which are based on stability property and weak convergence of BSDEs in the S-topology