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

Abstract : We establish an averaging principle for a family of solutions $(X^{\varepsilon}, Y^{\varepsilon})$ $:=$ $(X^{1,\,\varepsilon},\, X^{2,\,\varepsilon},\, Y^{\varepsilon})$ of a system of SDE-BSDE with a null recurrent fast component $X^{1,\,\varepsilon}$. In contrast to the classical periodic case, we can not rely on an invariant probability and the slow forward component $X^{2,\,\varepsilon}$ cannot be approximated by a diffusion process. On the other hand, we assume that the coefficients admit a limit in a \`{C}esaro sense. In such a case, the limit coefficients may have discontinuity. We show that we can approximate the triplet $(X^{1,\,\varepsilon},\, X^{2,\,\varepsilon},\, Y^{\varepsilon})$ by a system of SDE-BSDE $(X^1, X^2, Y)$ where $X := (X^1, X^2)$ 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 $W_{d+1,\text{loc}}^{1,2}(\R_+\times\R^d)$--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.
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https://hal-univ-tln.archives-ouvertes.fr/hal-01188449
Contributor : Khaled Bahlali <>
Submitted on : Sunday, August 30, 2015 - 12:59:17 PM
Last modification on : Tuesday, October 9, 2018 - 1:26:02 PM
Long-term archiving on : Wednesday, April 26, 2017 - 10:42:33 AM

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### Identifiers

• HAL Id : hal-01188449, version 1
• ARXIV : 1508.07696

### Citation

K Bahlali, A Elouaflin, E Pardoux. Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media. 2015. ⟨hal-01188449⟩

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