Averaging for BSDEs with null recurrent fast component. Application to homogenization in a non periodic media

We establish an averaging principle for a family of solutions (Xε,Yε):=(X1,ε,X2,ε,Yε) of a system of decoupled forward backward stochastic differential equations (SDE-BSDE for short) with a null recurrent fast component X1,ε. In contrast to the classical periodic case, we can not rely on an invariant probability and the slow forward component X2,ε cannot be approximated by a diffusion process. On the other hand, we assume that the coefficients admit a limit in a Cesàro sense. In such a case, the limit coefficients may have discontinuity. We show that the triplet (X1,ε,X2,ε,Yε) converges in law to the solution (X1,X2,Y) of a system of SDE-BSDE, where X:=(X1,X2) is a Markov diffusion which is the unique (in law) weak solution of the averaged forward component and Y is the unique solution to the averaged backward component. This is done with a backward component whose generator depends on the variable z. As application, we establish an homogenization result for semilinear PDEs when the coefficients can be neither periodic nor ergodic. We show that the averaged BDSE is related to the averaged PDE via a probabilistic representation of the (unique) Sobolev Wd+1,loc1,2(R+×Rd)-solution of the limit PDEs. Our approach combines PDE methods and probabilistic arguments which are based on stability property and weak convergence of BSDEs in the S-topology.