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

Abstract

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

Approximation of a degenerate semilinear PDEs with a nonlinear Neumann boundary condition

“…then we show that for each n, equation (3) has a viscosity solution u n which converges to a function u, and u is a viscosity solution to (2). Our method is probabilistic.…”

“…Usually, when the nonlinearity f depends in the gradient of the solution, PDEs techniques are used to control the gradient ∇u in order to get the convergence of the associated BSDE. And generally, a uniform ellipticity of the diffusion is required to get a good control of the gradient ∇u, see for instance [2,5,6] where this method is used in homogenization of nonlinear PDEs.…”

“…1,2 ([0, T ] × R d ), and (t, x) ∈ (0, T ] × D at which u i − ϕ has a local maximum, one has − ∂ϕ ∂t (t, x) − Lϕ(t, x) − f i (t, x, u(t, x), (∇ϕσ)(t, x)) ≤ 0, if x ∈ D, − h i (t, x, u(t, x)) ≤ 0, if x ∈ ∂D.…”

Probabilistic approach to singular perturbations of viscosity solutions to nonlinear parabolic PDEs

“…There is also a vast literature on the singular perturbations of nonlinear PDEs based on probabilistic argument. With the help of backward stochastic differential equations (BSDEs), Buckdahn and Hu [4] studied homogenization of viscosity solutions to semilinear parabolic PDEs with periodic structures, and Bahlali, Elouaflin abd Pardoux [2,3] extended the results of [26] to semilinear parabolic PDEs. In [6], Buckdahn and Ichihara considered homogenization of fully nonlinear parabolic PDEs in periodic case by stochastic control approach.…”

New characterizations of the $S$ topology on the Skorokhod space

“…Later the S topology was used in problems related to homogenization of stochastic differential equations (e.g. [1], [2], [5], [25], [30], [35], [32]), diffusion approximation of solutions to the Poisson equation ( [31]), stability of solutions to semilinear equations with Dirichlet operator ( [19]), martingale transport on the Skorokhod space ( [10]), the Skorokhod problem ( [23], [24], [28], [34]), econometrics ( [7]), control theory ( [3], [22]), linear models with heavy-tails ( [4]), continuity of semilinear Neumann-Dirichlet problems ( [27]), generalized Doob-Meyer decomposition ( [15]), modeling stochastic reaction networks ( [16]) and even in some considerations of more general character ( [8], [21]).…”