Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media
Résumé
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.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)