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We establish an averaging principle for a family ofsolutions$(X^{\varepsilon}, Y^{\varepsilon})$ $ :=$$(X^{1,\,\varepsilon},\,X^{2,\,\varepsilon},\, Y^{\varepsilon})$ of a system ofSDE-BSDEwith a null recurrent fast component $X^{1,\,\varepsilon}$. Incontrastto the classical periodic case, we can not rely on aninvariant probability andthe slow forward component$X^{2,\,\varepsilon}$ cannot be approximated by adiffusion process.On the other hand, we assume that the coefficients admit alimit in a\`{C}esaro sense. In such a case, the limit coefficients mayhavediscontinuity. We show that we can approximate thetriplet$(X^{1,\,\varepsilon},\, X^{2,\,\varepsilon},\, Y^{\varepsilon})$ byasystem of SDE-BSDE $(X^1, X^2, Y)$ where $X := (X^1, X^2)$ is aMarkov diffusionwhich is the unique (in law) weak solution of theaveraged forward component and$Y$ is the unique solution to the averaged backward component. This is donewith a backward component whosegenerator depends on the variable $z$.Asapplication, we establish an homogenization result for semilinearPDEs whenthe coefficients can be neither periodic nor ergodic. Weshow that the averagedBDSE is related to the averaged PDE via aprobabilistic representation of the(unique) Sobolev $W\_{d+1,\text{loc}}^{1,2}(\R\_+\times\R^d)$--solution of thelimitPDEs. Our approach combines PDE methods and probabilistic argumentswhichare based on stability property and weak convergence of BSDEsin the S-topology.

Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media