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-BSDEwith a null recurrent fast component $X^{1,\,\varepsilon}$. Incontrast
to the classical periodic case, we can not rely on aninvariant 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
havediscontinuity. We show that we can approximate the
triplet$(X^{1,\,\varepsilon},\, X^{2,\,\varepsilon},\, Y^{\varepsilon})$ bya
system of SDE-BSDE $(X^1, X^2, Y)$ where $X := (X^1, X^2)$ is aMarkov diffusion
which is the unique (in law) weak solution of theaveraged forward component and
$Y$ is the unique solution to the averaged backward component. This is done
with a backward component whosegenerator depends on the variable $z$.
Asapplication, we establish an homogenization result for semilinearPDEs when
the coefficients can be neither periodic nor ergodic. Weshow that the averaged
BDSE 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 the
limitPDEs. Our approach combines PDE methods and probabilistic argumentswhich
are based on stability property and weak convergence of BSDEsin the S-topology

Bahlali, K

Elouaflin, A

Pardoux, E

English

arXiv

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

Bahlali, Khaled

Elouaflin, Abouo

PARDOUX, Etienne

Archive Ouverte en Sciences de l'Information et de la Communication

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

Khaled Bahlali

Abouo Elouaflin

Etienne Pardoux

Crossref

Stochastic Processes and their Applications

International audienceWe 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-BSDEwith a null recurrent fast component $X^{1,\,\varepsilon}$. Incontrast to the classical periodic case, we can not rely on aninvariant 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 havediscontinuity. We show that we can approximate the triplet$(X^{1,\,\varepsilon},\,  X^{2,\,\varepsilon},\, Y^{\varepsilon})$ bya system of SDE-BSDE $(X^1, X^2, Y)$ where $X := (X^1, X^2)$ is aMarkov diffusion which is the unique (in law) weak  solution of theaveraged forward  component and $Y$ is the unique solution to the averaged backward component. This is done with a backward component whosegenerator depends on the variable $z$. Asapplication, we establish an homogenization result for semilinearPDEs when the coefficients can be neither periodic nor ergodic.  Weshow that the averaged BDSE 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 the limitPDEs. Our approach combines PDE methods and probabilistic argumentswhich are based on stability property and weak convergence of BSDEsin the S-topology

Elouaflin, Aboulo

Pardoux, Etienne

HAL AMU