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Nonlinear stochastic homogenization. (English) Zbl 0607.49010
In the framework of calculus of variations, the authors study stochastic homogenization of integral functionals. The integrand is supposed measurable in the first variable and convex in the second one. The main feature of this paper is to pass from the view-point of the stochastic differential equations to be solved to that one of the random integral functionals to be minimized. In order to study the random integral functionals, that are ”measurable” maps \(\omega\to F(\omega)\) from a probabilistic space \(\Omega\) into the functional class \({\mathcal F}\), and to study their convergence, a topological structure is given on \({\mathcal F}.\)
The applications of this result are concerned with a large number of real phenomena in physics, chemistry and engineering, where the structures to be homogenized are not periodic but only stochastically periodic.
Reviewer: M.Codegone

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49J55 Existence of optimal solutions to problems involving randomness
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
49K45 Optimality conditions for problems involving randomness
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