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Nonlinear stochastic homogenization and ergodic theory. (English) Zbl 0582.60034
Let \(\Omega\) be a probability space and let \(f: \Omega\) \(\times {\mathbb{R}}^ n\times {\mathbb{R}}^ n\to {\mathbb{R}}\) be a measurable function such that \(c_ 1| p|^{\alpha}\leq f(\omega,x,p)\leq c_ 2(1+| p|^{\alpha})\) where \(\alpha >1\) and \(c_ 2\geq c_ 1>0\) are fixed constants. Assume that f(\(\omega\),x,p) is convex in p and periodic in law with respect to x, i.e. the laws of the two vector random variables \((f(\cdot,x_ i,p_ i))_{i\in I}\) and \((f(\cdot x_ i+z,p_ i))_{i\in I}\) are equal for every \(z\in {\mathbb{Z}}^ n\) and for every finite family \(\{(x_ i,p_ i)\}_{i\in I}\) in \({\mathbb{R}}^ n\times {\mathbb{R}}^ n.\)
By using the subadditive ergodic theorem for spatial processes due to M. A. Akcoglu and U. Krengel [ibid. 323, 53-67 (1981; Zbl 0453.60039)], the authors prove that there exist a set \(\Omega\) ’\(\subseteq \Omega\) of full measure and a function \(f_ 0: \Omega \times {\mathbb{R}}^ n\to {\mathbb{R}}\), convex in p and satisfying the inequalities \(c_ 1| p|^{\alpha}\leq f_ 0(\omega,p)\leq c_ 2(1+| p|^{\alpha}),\) such that the sequence of minimum values of the problems \[ (P_{\epsilon})\quad \min_{u}\{\int_{A}f(\omega,x/\epsilon,Du(x))dx:\quad u-w\in W_ 0^{1,\alpha}(A)\} \] converges, as \(\epsilon\) \(\to 0\), to the minimum value of the problem \[ (P_ 0)\quad \min_{u}\{\int_{A}f_ 0(\omega,Du(x))dx:\quad u-w\in W_ 0^{1,\alpha}(A)\} \] for every \(\omega\in \Omega '\), for every bounded open subset A of \({\mathbb{R}}^ n\), and for every boundary value \(w\in W^{1,\alpha}(A)\). If, in addition, the random integrand f is ergodic, then f(\(\omega\),p) does not depend on \(\omega\).

MSC:
60F05 Central limit and other weak theorems
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