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Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients. (English) Zbl 1073.35021
The author considers the system of \(Q\) quasilinear equations (here \((\theta_{\epsilon})_l\) which depends on \((t,x) \in [0,T]\times {\mathbb R}^P\) and \(l=1, \dots Q\)) \( {\mathcal E}(\varepsilon)\): \[ \begin{split}{{\partial (\theta_{\varepsilon})_l}\over{\partial t}} +{{1}\over{2}} \sum_{i,j=1}^P a_{i,j}(\varepsilon^{-1}x, \theta_{\epsilon}) {{\partial^2 (\theta_{\varepsilon})_l}\over{\partial x_i \partial x_j}} + \sum_{i=1}^P \big[\varepsilon^{-1} b_{i}(\varepsilon^{-1}x, \theta_{\varepsilon}) + c_{i}(\varepsilon^{-1}x, \theta_{\varepsilon}, \nabla_x \theta_{\varepsilon})\big] {{\partial (\theta_{\varepsilon})_l}\over{\partial x_i}} \\+ \varepsilon^{-1} e_l(\varepsilon^{-1}x, \theta_{\epsilon}) + f_l(\varepsilon^{-1}x, \theta_{\epsilon}, \nabla_x \theta_{\epsilon}) = 0,\end{split} \] \[ \theta_{\varepsilon}(T,x) = H(x), \] where the coefficients \(a, b, c, e, f\) are \([0,1]^P\)-periodic functions and \(a\) is assumed to be symmetric. The aim is to study the behaviour of the system when \(\varepsilon\) goes to zero. To treat (from a probabilistic point of view) this system, the author represents the system by the forward-backward stochastic differential equations (here \(X_r, Y_r, Z_r\) depend on \((\varepsilon,t,x)\)) \(E(\varepsilon, t, x)\):
\[ \begin{aligned} X_s &= x + \int_t^s (\varepsilon^{-1} b + c) (\varepsilon^{-1}X_r, Y_r, Z_r) \,dr + \int_t^s \sigma (\varepsilon^{-1}X_r, Y_r) \,d B_r ,\\ Y_s &= H(X_T) + \int_s^T \varepsilon^{-1} e + f) (\varepsilon^{-1}X_r, Y_r, Z_r) \,dr - \int_s^T Z_r\sigma (\varepsilon^{-1}X_r, Y_r) \,d B_r ,\\ {\mathbf E}&\int_t^T(| X_s| ^2+| Y_s| ^2+| Z_s| ^2) \,ds < + \infty , \end{aligned} \] where \((\varepsilon^{-1} b + c)(x,y,z) = \varepsilon^{-1} b(x,y) + c(x,y,z)\) and \((\varepsilon^{-1} e + f)(x,y,z) = \varepsilon^{-1} e(x,y) + f(x,y,z)\), \((B_t)_{t\geqslant 0}\) is a Brownian motion and \(\sigma\) is such that \(\sigma \sigma^{\ast} = a\).
The connection between \({\mathcal E}(\varepsilon)\) and \(E(\varepsilon, t, x)\) can be roughly summarized as follows: for every \(s \in [t,T]\) \(Y_s(\varepsilon,t,x) = \theta_{\varepsilon}(s, X_s(\varepsilon,t,x))\). To study problems \(E(\varepsilon, t, x)\) the strategy is the following: pass to some modified processes (denoted by \(\widehat{X}\) and \(\widehat{Y}\)) in order to get rid of the terms \(\varepsilon^{-1} b\) and \(\varepsilon^{-1} e\), which leads to the study of the “auxiliary problems”; estimate the distance between \(\theta (\cdot , \widehat{X}(\varepsilon, t,x))\) and \(\widehat{Y}(\varepsilon, t,x)\), where \(\theta\) is the solution of the presumed limit system; to do this, since \(\theta\) is not regular enough, the author needs to pass through a regularization of \(\theta\), an appropriate sequence \((\zeta_n)_n\), and introduce some “auxiliary SDEs”, a new approach needed to by-pass a problem due to \((\zeta_n)_n\) (a control, uniform in \(n\), of \(D_{xx}\zeta_n\)). Finally the author establishes that \[ \begin{split} \lim_{\varepsilon \to 0} \biggl( {\mathbf E} \sup_{t \leqslant s \leqslant T} | Y_s(\varepsilon, t,x) - \theta (s, X_s(\varepsilon, t,x))| ^2 + {\mathbf E} \int_t^T | Z_s(\varepsilon, t,x) - \nabla_x \theta(s,X_s(\varepsilon, t,x)) \\ \chi_1(\varepsilon^{-1}X_s(\varepsilon, t,x),Y_s(\varepsilon, t,x)) - \chi_2(\varepsilon^{-1}X_s(\varepsilon, t,x),Y_s(\varepsilon, t,x)) | ^2\,ds\biggr) = 0 \end{split} \] where \(\chi_1\) and \(\chi_2\) are appropriate corrector terms.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K55 Nonlinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
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