Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices

A homogenization problem for infinite dimensional diffusion processes indexed by ${\mathbf{Z}}^d$ having periodic drift coefficients is considered. By an application of the uniform ergodic theorem for the infinite dimensional diffusion processes based on logarithmic Sobolev inequalities, an $L^1$ type homogenization property of the processes with respect to an invariant measure is proved. This is the, so far, best possible analogue in infinite dimensions to a known result in the finite dimensional case (cf. [G. Papanicolaou, S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979. [4]]). To cite this article: S. Albeverio et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).