Stochastic homogenization and random lattices

We present some variants of stochastic homogenization theory for scalar elliptic equations of the form - [GRAPHICS] These variants basically consist in defining stochastic coefficients [GRAPHICS] from stochastic deformations (using random diffeomorphisms) of the periodic setting, as announced in [X. Blanc, C. Le Bris, P.-L. Lions, Une variante de la theorie de l'homogeneisation stochastique des operateurs elliptiques (A variant of stochastic homogenization theory for elliptic operators), C. R. Acad. Sci. Ser. 1343 (2006) 717-727]. The settings we define are not covered by the existing theories. We also clarify the relation between this type of questions and our construction, performed in [X. Blanc, C. Le Bris, P.-L. Lions, A definition of the ground state energy for systems composed of infinitely many particles, Commun. Partial Differential Equations 28 (1-2) (2003) 439-475; X. Blanc, C. Le Bris, P.-L. Lions, The energy of some microscopic stochastic lattices, Arch. Rat. Mech. Anal. 184 (2) (2007) 303-339], of the energy of, both deterministic and stochastic, microscopic infinite sets of points in interaction. (c) 2007 Elsevier Masson SAS. All rights reserved.