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Large deviations, central limit theorems and $L^p$ convergence for Young measures and stochastic homogenizations

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  • Mathematics


mpiau.dvi ESAIM� Probability and Statistics November ����� Vol� �� � ���� LARGE DEVIATIONS� CENTRAL LIMIT THEOREMS AND L p CONVERGENCE FOR YOUNG MEASURES AND STOCHASTIC HOMOGENIZATIONS JULIEN MICHEL AND DIDIER PIAU Abstract� We study the stochastic homogenization processes consid� ered by Baldi ������ and by Facchinetti and Russo ������ We precise the speed of convergence towards the homogenized state by proving the following results �i� a large deviations principle holds for the Young measures� if the Young measures are evaluated on a given function� then �ii� the speed of convergence is bounded in every L p norm by an explicit rate and �iii� central limit theorems hold In dimension �� we apply these results to the stochastic homogenization of random p�Laplacian operators for any p � � �� Introduction A non homogeneous material lies in a topological space � and its prop� erties at a point x are described by the element a�x� of a topological space Z� For instance� the function a � � � Z may be the thermic conductivity of the material� its electric resistivity� or a deterministic function of these quantities� If the scale of the irregularities of the material is small� the func� tion a is highly non regular� One can guess that the full knowledge of a� in other words a complete microscopic description of the material� is at the same time impossible to get and irrelevant if one is interested only in the macroscopic properties of the material� One way of avoiding this problem is to replace a by suitable random functions a � ��� �� � �� Z where � is the alea and � is the typical scale of the irregularities of a � � When � goes to zero� one expects that the behaviour of the random ap� proximated material which is described by a � ��� �� converges� in a sense� to the behaviour of the actual material� This mean description� often called stochastic homogenization� is intensely studied in the physical and mathe� mati

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