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

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