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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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