2015/08/31 by Khaled Bahlali, Bahlali, K, Abouo Elouaflin +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1508.07696
openalex publication_date 2015/08/31 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We establish an averaging principle for a family of solutions(Xε, Yε) := (X1, ε, X2, ε, Yε) of a system of SDE-BSDEwith a null recurrent fast component X1, ε. Incontrast to the classical periodic case, we can not rely on aninvariant probability and the slow forward componentX2, ε cannot be approximated by a diffusion process.On the other hand, we assume that the coefficients admit a limit in aCesaro sense. In such a case, the limit coefficients may havediscontinuity. We show that we can approximate the triplet(X1, ε, X2, ε, Yε) bya system of SDE-BSDE (X1, X2, Y) where X := (X1, X2) 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,loc1,2(\R_+×\Rd)--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.