Homogenization of semilinear PDEs with discontinuous averaged coefficients

Abstract

We study the asymptotic behavior of the solution of semi-linear PDEs. Neither periodicity nor ergodicity assumptions are assumed. The coefficients admit only a limit in aCesaro sense. In such a case, the limit coefficients may have discontinuity. We use probabilistic approach based on weak convergence techniques for the associated backward stochastic differential equation in the S-topology. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of L p -viscosity solution.

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

“…Indeed, building on the PDEs, we construct a sequence of semimartingales (Z ε,n ) that we substitute to (Z ε ). This allows us to use the method developed in [1,2,23]. Next, we show that the problems with (Z ε,n ) and that with (Z ε ) average to the same limit.…”

“…The first one plays a similar role to that played by the invariant measure in the periodic case. It was introduced in [23] for a forward SDE and later adapted in [1] to systems of SDE-BSDE in which the generator of the backward component does not depend on the variable Z. We do not provide a proof, since that of Lemma 4.7 in [1] can be repeated word to word (also we have a new variable).…”

“…In contrast to these two classical situations (deterministic periodic and random stationary coefficients) which were mainly studied, we consider in this paper a different situation, building upon earlier results of [23] and more recently those of [1,2]. We extend the results of [23] to systems of SDE-BSDEs and those of [1,2] to the case where the generator f of the BSDE component depends upon the second unknown of the BSDE. As a consequence, we derive an homogenization result for semilinear PDEs when the nonlinear part depends on the solution as well as on its gradient.…”

“…However, in [1,2] the backward equation does not depend on the control variable. More precisely, the result of [23] was extended, in [1,2], to the following SDE-BSDE. where M X x,ε is the martingale part of the process X x,ε := (X 1,x,ε , X 2,x,ε ).…”

“…Therefore, we will also study the asymptotic behavior of the PDE (1.6). As in [1,2,23], we consider here the averaged coefficients as limits in the Cesàro sense. Usually, the averaged coefficients are computed as means with respected to the (unique) invariant probability measure.…”